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Chapter 2: Problem 20

Find the rational zeros of the function. $$f(x)=x^{3}-13 x+12$$

Short Answer

Expert verified
The rational zero of the function \(f(x) = x^3 - 13x + 12\) is \(x = 3\).

Step by step solution

01

Identify Potential Rational Zeros

By the Rational Root Theorem, the potential rational zeros of a polynomial function are the ratio of the factors of the constant term to the factors of the leading coefficient. In this case, the constant term is 12 and the leading coefficient is 1. The factors of 12 are 卤1, 卤2, 卤3, 卤4, 卤6, and 卤12. Since the leading coefficient is one, the potential rational zeros are 卤1, 卤2, 卤3, 卤4, 卤6, and 卤12.
02

Test Potential Zeros

We substitute each potential zero, p, into the function to check if \(f(p) = 0\). Any p for which \(f(p) = 0\) is a rational zero. After substituting, we find \(f(1) = -13 + 12 = -1\), \(f(-1) = 13 + 12 = 25\), \(f(2) = 8 - 26 + 12 = -6\), \(f(-2) = -8 - 26 + 12 = -22\), \(f(3) = 27 - 39 + 12 = 0\), \(f(-3) = -27 - 39 + 12 = -54\), and we can see \(f(4)\), \(f(-4)\), \(f(6)\), and \(f(-6)\) are not zero, and \(f(12)\), \(f(-12)\) are not zero either. So we only found one rational zero which is 3.
03

Conclude Rational Zeros

The only valid rational zero we found for the function \(f(x) = x^3 - 13x + 12\) is 3. Therefore, the function has one rational zero, x = 3.

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