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Chapter 2: Problem 35

Use Descartes' Rule of Signs to state the number of possible positive and negative real zeros of each polynomial function. $$P(x)=x^{5}-32$$

Short Answer

Expert verified
Based on Descartes' Rule of Signs, the polynomial function \(P(x) = x^{5}-32\) has exactly one positive real root and no negative real roots.

Step by step solution

01

Identify the coefficients of the polynomial function

The polynomial is \(P(x) = x^{5} - 32\). It has two terms. The coefficients of these terms are 1 and -32, respectively. The others are zero coefficients for \(x^{4}\), \(x^{3}\), \(x^{2}\), and \(x\).
02

Apply Descartes' Rule of Signs for positive real zeroes

For positive real zeros, we simply need to count the number of sign changes in the coefficients: 1, 0, 0, 0, 0, and -32. There is only one sign change. So, the given polynomial function has exactly one positive real root.
03

Apply Descartes' Rule of Signs for negative real zeroes

To find the number of negative real zeroes, replace 'x' with '-x' in the polynomial. We get \(P(-x) = (-x)^{5} - 32\), which simplifies to \(-x^{5} - 32\). The coefficients are -1, 0, 0, 0, 0, and -32. There isn't any change in sign. Hence, the polynomial function has no negative root.

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