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Chapter 2: Problem 7

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

Short Answer

Expert verified
The potential rational zeros for the polynomial function \(P(x)=x^{3}+3 x^{2}-6 x-8\) using the Rational Zero Theorem are 卤1, 卤2, 卤4, 卤8.

Step by step solution

01

Identify the constant term and leading coefficient

The constant term in the polynomial function \(P(x)=x^{3}+3 x^{2}-6 x-8\) is -8 and the leading coefficient is 1.
02

Find the factors of the constant term and leading coefficient.

The factors of -8 (constant term) are 卤1, 卤2, 卤4, 卤8. The factors of 1 (leading coefficient) are 卤1.
03

Use the Rational Zero Theorem to list the possible rational zeros.

Using the Rational Zero Theorem, the possible rational zeros are given by 卤 p/ q, where p are factors of the constant term and q are the factors of the leading coefficient. Thus, the possible rational zeros for the given polynomial function are every possible combination of \(卤(1, 2, 4, 8)/1\), that is 卤1, 卤2, 卤4, 卤8.

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