Question

1. which set of numbers for x might be tested to determine the possible integral zeroes of $x^{4}-2x^{3}-7x^{2}-8x + 12$? a. $pm{1,2,4,12}$ b. $pm{1,2,3,4,6}$ c. $pm{1,2,3,4,6,8}$ d. $pm{1,2,3,4,6,12}$

Explanation:

Step1: Apply Rational Root Theorem

The Rational Root Theorem states that if a polynomial \(a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\) has integer roots, they are of the form \(\frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading - coefficient \(a_n\). For the polynomial \(x^{4}-2x^{3}-7x^{2}-8x + 12\), \(a_n = 1\) and \(a_0=12\).

Step2: Find factors of \(a_0\)

The factors of the constant term \(a_0 = 12\) are \(p=\pm1,\pm2,\pm3,\pm4,\pm6,\pm12\). Since \(q = 1\) (because the leading - coefficient \(a_n = 1\)), the possible integral zeroes are the factors of \(12\).

Answer:

D. \(\pm\{1,2,3,4,6,12\}\)