# How do you use the Intermediate Value Theorem to show that the polynomial function # x^3+2x^2-42# has a root in the interval [0, 3]?

This function is negative at

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To use the Intermediate Value Theorem to show that the polynomial function (x^3 + 2x^2 - 42) has a root in the interval ([0, 3]), you need to demonstrate that the function changes sign over that interval. First, evaluate the function at the endpoints of the interval: (f(0)) and (f(3)). If the values have different signs, then there exists at least one root of the function in the interval ([0, 3]).

(f(0) = 0^3 + 2(0)^2 - 42 = -42)

(f(3) = 3^3 + 2(3)^2 - 42 = 27 + 18 - 42 = 3)

Since (f(0) = -42) and (f(3) = 3), and (f(0)) is negative while (f(3)) is positive, the function changes sign over the interval ([0, 3]). Thus, by the Intermediate Value Theorem, there exists at least one root of the function (x^3 + 2x^2 - 42) in the interval ([0, 3]).

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