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Chapter 2: Problem 7

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

Short Answer

Expert verified
The possible rational zeros of \( f(x) \) are \( \pm1, \pm2, \pm3, \pm4, \pm6, \pm12 \)

Step by step solution

01

Identify the Constant Term and Leading Coefficient

In the given polynomial function, the constant term is \( -12 \) and the leading coefficient is \( 1 \) (the coefficient of the term \( x^{5} \)).
02

Factorize the Constant and the Leading Coefficient

The factors of \( -12 \) are \( \pm1, \pm2, \pm3, \pm4, \pm6, \pm12 \). Since the leading coefficient is \( 1 \), the factors of the leading coefficient are \( \pm1 \).
03

Form All Possible Combinations

The Rational Zero Theorem states that we can obtain potential rational zeros by forming fractions from the factors of the constant term and factors of the leading coefficient. Since the factors of the leading coefficient are \( \pm1 \), the potential rational zeros of the polynomial are same as the factors of \( -12 \), which are \( \pm1, \pm2, \pm3, \pm4, \pm6, \pm12 \).

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Most popular questions from this chapter

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-1}{x}$$

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