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Chapter 2: Problem 15

Use the Rational Zero Theorem to list possible rational zeros for each polynomial function. $$P(x)=x^{5}-32$$

Short Answer

Expert verified
The possible rational zeros of the polynomial function \(P(x)=x^{5}-32\) are -32, -16, -8, -4, -2, -1, 1, 2, 4, 8, 16, and 32.

Step by step solution

01

Identify coefficients

Find the coefficients in the polynomial function. For the function \(P(x)=x^{5}-32\), the constant term (coefficient without \(x\)) is -32 and the leading coefficient (coefficient of the highest-order term) is 1.
02

Determine factors

Determine all the factors of the constant term and the leading coefficient. Factors of -32 are -1, 1, -2, 2, -4, 4, -8, 8, -16, 16, -32, and 32. The factors of 1 are -1 and 1.
03

List of possible rational zeros

Create the list of possible rational zeros by dividing each factor of the constant term by each factor of the leading coefficient. In this case, since the leading coefficient is 1, our possible rational zeros are the factors of the constant term itself: -32, -16, -8, -4, -2, -1, 1, 2, 4, 8, 16, and 32

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