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Chapter 2: Problem 10

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

Short Answer

Expert verified
The possible rational zeros of the given polynomial are: 卤1/3, 卤2/3, 卤4/3, 卤8/3, 卤1, 卤2, 卤4, and 卤8.

Step by step solution

01

Identify the Coefficients

Firstly, identify the coefficients of the polynomial. In this case, for \(P(x)=3 x^{3}+11 x^{2}-6 x-8\), the constant term (a0) is -8, and the leading coefficient (an) is 3.
02

Find Factors

To proceed, find the factors for the constant term (a0 = -8) and the leading coefficient (an = 3). The factors of -8 are -1, 1, -2, 2, -4, 4, -8, 8. And for the leading coefficient 3, the factors are -1, 1, -3, 3, which potential divisors of any rational root of the polynomial according to the Rational Zero theorem.
03

List Possible Zeros

The possible rational zeros of the polynomial are determined by \(\frac{p}{q}\), where p are factors of the constant term (a0 = -8) and q are factors of the leading coefficient (an = 3). Therefore, the possible rational zeros are: 卤1/3, 卤2/3, 卤4/3, 卤8/3, 卤1, 卤2, 卤4, and 卤8.

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Most popular questions from this chapter

Given \(g(x)=4 x^{4}-6 x^{2}+5,\) find \(g(-2) .[1.3]\)

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