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Chapter 2: Problem 105

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Short Answer

Expert verified
The rational zeros of the function are 1, -1/2, and -1/4.

Step by step solution

01

Apply the Rational Root Theorem

Apply the Rational Root theorem to list all possible rational roots of the function. The last term (the constant) is 1 and the coefficient of the leading term (the highest power of \(x\)) is 4. The factors of 1 are 卤1 and the factors of 4 are 卤1, 卤2, and 卤4. So, the possible rational roots are 卤1/1, 卤1/2, 卤1/4, 卤2/1, 卤2/2 and 卤2/4, which simplify to 卤1, 卤1/2, 卤1/4, 卤2, 卤1 and 卤1/2. Thus, the possible rational roots of the polynomial function are 卤1, 卤1/2 and 卤1/4.
02

Test each possible rational root

Next, test each possible rational root in the polynomial equation to see which ones are actually roots. To do this, substitute each possible root for \(x\) in the equation \(f(x)=0\) and see if it holds true.
03

Find the rational roots

By substituting each possible rational root into the polynomial function, it is found that 1, -1/2, and -1/4 are the roots of the function. Thus, the rational zeros of the function are 1, -1/2, and -1/4.

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Most popular questions from this chapter

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=x^{3}-4 x^{2}+1\) (a) Upper: \(x=4\) (b) Lower: \(x=-1\)

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Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{4}-4 x^{3}+8 x^{2}-16 x+16$$

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