Question

what are the simplest factors of this polynomial?
x³ + 4x² - 9x - 36
a. (x - 4)(x² + 3)
b. (x + 4)(x² + 3)
c. (x + 4)(x + 3)(x - 3)
d. (x - 4)(x + 3)(x - 3)

Explanation:

Step1: Group the polynomial

Group the terms of \(x^{3}+4x^{2}-9x - 36\) as \((x^{3}+4x^{2})+(-9x - 36)\).

Step2: Factor out common terms

Factor out \(x^{2}\) from the first group and \(-9\) from the second group: \(x^{2}(x + 4)-9(x + 4)\).

Step3: Factor out \((x + 4)\)

Now, factor out \((x + 4)\) to get \((x + 4)(x^{2}-9)\).

Step4: Factor the difference of squares

Notice that \(x^{2}-9\) is a difference of squares, which factors as \((x + 3)(x - 3)\). So the fully factored form is \((x + 4)(x + 3)(x - 3)\).

Answer:

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