Question

question 4 of 10
express the polynomial as a product of linear factors.
$f(x) = x^3 - 5x^2 - 18x + 72$

a. $(x - 3)(x + 3)(x - 8)$
b. $(x - 2)(x + 6)(x - 6)$
c. $(x - 3)(x + 4)(x - 6)$
d. $(x - 4)(x + 3)(x + 2)$

Explanation:

Step1: Test rational roots

By Rational Root Theorem, test \(x=3\): \(3^3 -5(3)^2 -18(3)+72=27-45-54+72=0\). So \(x-3\) is a factor.

Step2: Perform polynomial division

Divide \(x^3 -5x^2 -18x +72\) by \(x-3\):
\(x^3 -5x^2 -18x +72=(x-3)(x^2 -2x -24)\)

Step3: Factor quadratic

Factor \(x^2 -2x -24=(x+4)(x-6)\)

Step4: Combine factors

\(f(x)=(x-3)(x+4)(x-6)\)

Answer:

C. (x - 3)(x + 4)(x - 6)