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Chapter 2: Problem 6

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$f(x)=3 x^{4}-11 x^{3}-3 x^{2}-6 x+8$$

Short Answer

Expert verified
The list of all possible rational zeros is: \(\pm1, \pm2, \pm4, \pm8, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{8}{3}\).

Step by step solution

01

Identify the constant term and the leading coefficient

In the given function \(f(x) = 3x^{4}-11x^{3}-3x^{2}-6x+8\), the constant term is 8 and the leading coefficient is 3.
02

Find the factors of the constant term and the leading coefficient

The factors of 8 are \(\pm1, \pm2, \pm4, \pm8\) and the factors of 3 are \(\pm1, \pm3\).
03

List all possible rational zeros

Use the Rational Zero Theorem to list all possible rational zeros. These will be all combinations of the factors of 8 divided by the factors of 3. So, the possible rational zeros are \(\pm\frac{1}{1}, \pm\frac{2}{1}, \pm\frac{4}{1}, \pm\frac{8}{1}, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{8}{3}\). This simplifies to \(\pm1, \pm2, \pm4, \pm8, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{8}{3}\)

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