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Chapter 2: Problem 11

Use the Rational Zero Theorem to list possible rational zeros for each polynomial function. $$P(x)=6 x^{4}+23 x^{3}+19 x^{2}-8 x-4$$

Short Answer

Expert verified
The possible rational zeros for the polynomial function \(P(x)=6x^4+23x^3+19x^2-8x-4\) are: 卤1, 卤2, 卤4, 卤1/2, 卤1/3, 卤2/3, 卤4/3, 卤1/6.

Step by step solution

01

Identify the Coefficients

First, identify the constant term and the leading coefficient in the polynomial function. In the given polynomial \(P(x)=6x^4+23x^3+19x^2-8x-4\), the constant term is '-4' and the leading coefficient is '6'.
02

List the Factors

List all the factors of the constant term and the leading coefficient. The factors of '-4' are 卤1, 卤2, 卤4. The factors of '6' are 卤1, 卤2, 卤3, 卤6.
03

Apply the Rational Zero Theorem

Use the Rational Zero Theorem to list the possible rational zeros. According to the theorem, the possible rational zeros are all the ratios of 'p' (factors of the constant term) to 'q' (factors of the leading coefficient). Calculate these ratios to find the possible rational zeros.
04

Calculate the Ratios

Calculate the ratios by dividing each factor of the constant term by each factor of the leading coefficient. The possible rational zeros are: 卤1/1, 卤2/1, 卤4/1, 卤1/2, 卤2/2, 卤4/2, 卤1/3, 卤2/3, 卤4/3, 卤1/6, 卤2/6, 卤4/6. Simplify these ratios to get the final list of possible rational zeros.
05

Simplify the Ratios

Simplify the ratios calculated in Step 4. The list of possible rational zeros for the polynomial function then becomes: 卤1, 卤2, 卤4, 卤1/2, 卤2 (= 卤1), 卤2, 卤1/3, 卤2/3, 卤4/3, 卤1/6, 卤1/3 (= 卤2/6), 卤2/3. Eliminate any duplicates in this list.
06

Final List of Possible Rational Zeros

Eliminate any duplicates from the list obtained in Step 5. The final list of possible rational zeros for the polynomial function is: 卤1, 卤2, 卤4, 卤1/2, 卤1/3, 卤2/3, 卤4/3, 卤1/6.

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