# How do you use the Intermediate Value Theorem to show that the polynomial function #f(x) = x^4 + 8x^3 - x^2 + 2# has a zero in the interval [-1, 1]?

See the explanation.

By signing up, you agree to our Terms of Service and Privacy Policy

To use theTo use the IntermediateTo use the Intermediate Value Theorem toTo use the Intermediate Value Theorem to showTo use the Intermediate Value Theorem to show thatTo use the Intermediate Value Theorem to show that theTo use the Intermediate Value Theorem to show that the polynomialTo use the Intermediate Value Theorem to show that the polynomial functionTo use the Intermediate Value Theorem to show that the polynomial function (To use the Intermediate Value Theorem to show that the polynomial function ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(xTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = xTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8xTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - xTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 )To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) hasTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has aTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zeroTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero inTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in theTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the intervalTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]\To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]),To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), weTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we needTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the functionTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need toTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changesTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to showTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes signTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign overTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign withinTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over thatTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within thatTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that intervalTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that intervalTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

FirstTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluateTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

FirstTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) )To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) andTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) )To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) andTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 +To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

SinceTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

SinceTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 )To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) andTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 )To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) andTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ),To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), andTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and theTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ),To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the functionTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and theTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function isTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the functionTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuousTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 andTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 andTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by theTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, byTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate ValueTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by theTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value TheTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate ValueTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value TheTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, thereTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value TheoremTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there existsTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists aTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, thereTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a valueTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists atTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( cTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at leastTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) inTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( cTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in theTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c )To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) inTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]\To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1])To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]\To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) suchTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1])To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such thatTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) suchTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such thatTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that (To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(cTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( fTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(cTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c)To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) =To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial functionTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function hasTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has aTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 \To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zeroTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ).To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero inTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). ThusTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in theTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero inTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the intervalTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the intervalTo use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the interval ([-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the interval ([-To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the interval ([-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the interval ([-1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the interval ([-1, To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the interval ([-1,To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the interval ([-1, 1To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the interval ([-1, To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the interval ([-1, 1]\To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the interval ([-1, 1]\To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the interval ([-1, 1]).To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the interval ([-1, 1]).To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), we need to show that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = 1^4 + 8(1)^3 - 1^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists a value ( c ) in the interval ([-1, 1]) such that ( f(c) = 0 ). Therefore, the polynomial function has a zero in the interval ([-1, 1]).To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^4 + 8x^3 - x^2 + 2 ) has a zero in the interval ([-1, 1]), you need to demonstrate that the function changes sign within that interval.

First, evaluate ( f(-1) ) and ( f(1) ).

( f(-1) = (-1)^4 + 8(-1)^3 - (-1)^2 + 2 = 1 - 8 - 1 + 2 = -6 )

( f(1) = (1)^4 + 8(1)^3 - (1)^2 + 2 = 1 + 8 - 1 + 2 = 10 )

Since ( f(-1) = -6 ) and ( f(1) = 10 ), and the function is continuous between -1 and 1, by the Intermediate Value Theorem, there exists at least one ( c ) in ([-1, 1]) such that ( f(c) = 0 ). Thus, the polynomial function has a zero in the interval ([-1, 1]).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find all extrema for #f(x) = 2x + ln x#?
- What are the local extrema, if any, of #f (x) =(lnx)^2/x#?
- What are the extrema and saddle points of #f(x,y) = xy(e^(y^2)-e^(x^2))#?
- How do you verify that the function #f(x)=x/(x+6)# satisfies the hypotheses of The Mean Value Theorem on the given interval [0,1] and then find the number c that satisfy the conclusion of The Mean Value Theorem?
- Is #f(x)=-2x^2-2x-1# increasing or decreasing at #x=-1#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7