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Chapter 2: Problem 16

Use the Rational Zero Theorem to list possible rational zeros for each polynomial function. $$P(x)=x^{4}-1$$

Short Answer

Expert verified
The possible rational zeros for the polynomial function \(P(x) = x^{4} - 1\) are -1 and 1.

Step by step solution

01

Identifying the Coefficients

Given the polynomial function \(P(x) = x^{4} - 1\). The constant term (or the constant coefficient) here is -1 and the leading coefficient (the coefficient of the term with the highest power) is 1.
02

Determining Factors of the Coefficients

Identify the factors of the constant and leading coefficients. The factors of the constant term -1 are -1 and 1. Similarly, the factors of the leading coefficient 1 are -1 and 1.
03

Applying the Rational Zero Theorem

Use the Rational Zero Theorem which suggests that every rational zero is of the form \(\frac{p}{q}\), where \(p\) denotes any factor of the constant term (-1 in this case), and \(q\) represents any factor of the leading coefficient (1 in this case). Because both constant and leading coefficients (i.e., -1 and 1) have the same factors (-1 and 1), the set of possible rational zeros here consists of \(\frac{-1}{-1}\), \(\frac{-1}{1}\), \(\frac{1}{-1}\), and \(\frac{1}{1}\), simplifying to 1, -1, -1, and 1.
04

Listing the unique rational zeros

Collect all the different rational zeros from the previous step. Eliminate any duplicates in the list. Therefore, the possible rational zeros for the polynomial function \(P(x) = x^{4} - 1\) are -1 and 1.

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