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Chapter 2: Problem 4

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$f(x)=2 x^{4}+3 x^{3}-11 x^{2}-9 x+15$$

Short Answer

Expert verified
Applying the Rational Zero Theorem for the polynomial, the possible rational zeros are 卤1, 卤1/2, 卤3, 卤3/2, 卤5, 卤5/2, 卤15, 卤15/2.

Step by step solution

01

Identify Coefficients & Factors

Firstly, recognize the leading coefficient and the constant term from the function \(f(x)=2x^4+3x^3-11x^2-9x+15\). Here, the constant term is 15 and the leading coefficient is 2. The factors of 15 are 卤1, 卤3, 卤5, 卤15 and the factors of 2 are 卤1, 卤2.
02

Listing Possible Rational Zeros

The Rational Zero Theorem can then be applied to find all possible rational zeros of the function. It states that all possible rational zeros are \(p/q\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. Using the factors of 15 and 2, all possible rational zeros of the function could be 卤1, 卤1/2, 卤3, 卤3/2, 卤5, 卤5/2, 卤15, 卤15/2.

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