# How do you use the Intermediate Value Theorem to show that the polynomial function #P(x) = x^4 + 2x^3 + 2x^2 - 5x + 3# has a zero in the interval [0, 1]?

You can't.

Here is the graph:

graph{y=x^4+2x^3+2x^2-5x+3 [-27.77, 37.2, -11.18, 21.25]}

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To use the Intermediate Value Theorem to show that the polynomial function ( P(x) = x^4 + 2x^3 + 2x^2 - 5x + 3 ) has a zero in the interval ([0, 1]), you need to demonstrate that the function changes sign within that interval. Evaluate ( P(0) ) and ( P(1) ). If ( P(0) ) and ( P(1) ) have opposite signs, then by the Intermediate Value Theorem, there exists at least one zero of ( P(x) ) between ( 0 ) and ( 1 ).

Calculate ( P(0) ) and ( P(1) ) as follows:

( P(0) = (0)^4 + 2(0)^3 + 2(0)^2 - 5(0) + 3 = 3 )

( P(1) = (1)^4 + 2(1)^3 + 2(1)^2 - 5(1) + 3 = 3 )

Since ( P(0) = P(1) = 3 ), ( P(x) ) does not change sign on the interval ([0, 1]). Therefore, the Intermediate Value Theorem cannot be applied directly to show that ( P(x) ) has a zero in that interval.

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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.

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