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Chapter 2: Problem 12

Use the Rational Zero Theorem to list possible rational zeros for each polynomial function. $$P(x)=2 x^{3}+9 x^{2}-2 x-9$$

Short Answer

Expert verified
The possible rational zeros of the function \(P(x)=2 x^{3}+9 x^{2}-2 x-9\) are 卤1, 卤3, 卤9, 卤0.5, 卤1.5, and 卤4.5.

Step by step solution

01

Identify the Coefficients

First, identify the leading coefficient and the constant term. In the polynomial \(P(x)=2 x^{3}+9 x^{2}-2 x-9\), the leading coefficient is 2 and the constant term is -9.
02

Find Factors

Next, find all factors of the leading coefficient (2) and the constant term (-9). The factors of 2 are 卤1, 卤2 and factors of -9 are 卤1, 卤3, 卤9.
03

Create Rational Combinations

Now, we form all possible rational numbers 'a/b' using these factors. The potential rational zeros are 卤1/1, 卤3/1, 卤9/1, 卤1/2, 卤3/2, 卤9/2. Simplify these numbers to get the list: 卤1, 卤3, 卤9, 卤0.5, 卤1.5, 卤4.5.

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