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Chapter 2: Problem 14

Use the Rational Zero Theorem to list possible rational zeros for each polynomial function. $$P(x)=x^{5}-x^{4}-7 x^{3}+7 x^{2}-12 x-12$$

Short Answer

Expert verified
The possible rational zeros of the polynomial \(P(x) = x^{5} - x^{4} - 7x^{3} + 7x^{2} - 12x - 12\) are -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12.

Step by step solution

01

Identify Coefficients

The given polynomial function is \(P(x) = x^{5} - x^{4} - 7x^{3} + 7x^{2} - 12x - 12\). The leading coefficient is the coefficient of the highest degree term, which is 1 here (from \(x^{5}\)). The trailing coefficient, also called the constant term, is the coefficient with no variable attached, which is -12 here.
02

List Factors

We need to list all the factors (both positive and negative) of the leading and the trailing coefficient. The factors of 1 are 1 and -1. The factors of -12 are -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, and 12.
03

Apply Rational Zero Theorem

According to the Rational Zero Theorem, every rational zero of the polynomial is of the form \(p/q\), where \(p\) is a factor of the trailing coefficient and \(q\) is a factor of the leading coefficient. Therefore, we divide every factor of the trailing coefficient by every factor of the leading coefficient. In this case, the potential zeros of the polynomial are -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12.

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