Question

use the rational zero theorem to list all possible rational zeros for the given function. f(x)=x^3 - 7x^2 - 11x + 14. choose the answer below that lists all possible rational zeros. a. -1,1, -2,2, -7,7, -14,14 b. -1,1, -2,2, -7,7, -14,14, -\frac{1}{2},\frac{1}{2}, -\frac{1}{7},\frac{1}{7}, -\frac{1}{14},\frac{1}{14} c. -1,1, -14,14 d. -1,1, -\frac{1}{2},\frac{1}{2}, -\frac{1}{7},\frac{1}{7}, -\frac{1}{14},\frac{1}{14}

Explanation:

Step1: Identify the constant and leading - coefficient

The constant term of the polynomial \(f(x)=x^{3}-7x^{2}-11x + 14\) is \(p = 14\), and the leading - coefficient is \(q=1\).

Step2: Find the factors of \(p\) and \(q\)

The factors of \(p = 14\) are \(\pm1,\pm2,\pm7,\pm14\), and the factors of \(q = 1\) are \(\pm1\).

Step3: Apply the Rational Zero Theorem

The possible rational zeros are of the form \(\frac{p}{q}\). Since \(q = 1\), the possible rational zeros are the factors of \(14\), which are \(\pm1,\pm2,\pm7,\pm14\).

Answer:

A. \(-1,1,-2,2,-7,7,-14,14\)