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Chapter 2: Problem 2

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$f(x)=x^{3}+3 x^{2}-6 x-8$$

Short Answer

Expert verified
The possible rational zeros of the function \(f(x)=x^{3}+3 x^{2}-6 x-8\) by the Rational Zero Theorem are \(\pm1, \pm2, \pm4, \pm8\).

Step by step solution

01

Identify coefficients

The given polynomial function is \(f(x)=x^{3}+3 x^{2}-6 x-8\). Here, the leading coefficient \(a_{n}\) is 1 and the constant term \(a_{0}\) is -8.
02

Calculate factors

Next, identify the factors of \(a_{n}\) and \(a_{0}\). The only factor of 1 is 1 itself. The factors of -8 are \(\pm1, \pm2, \pm4, \pm8\).
03

Formulate possible rational zeros

Combine the factors of \(a_{n}\) and \(a_{0}\) to obtain all reasonable rational zeros by dividing each factor of -8 by the factor of 1. The possible rational zeros are therefore \(\pm1, \pm2, \pm4, \pm8\).

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