if $f(-2)=0$, what are all the factors of the function $f(x)=x^{3}-2x^{2}-68x - 120$? use the remainder…

if $f(-2)=0$, what are all the factors of the function $f(x)=x^{3}-2x^{2}-68x - 120$? use the remainder theorem.\n$(x + 2)(x + 60)$\n$(x - 2)(x - 60)$\n$(x - 10)(x + 2)(x + 6)$\n$(x + 10)(x - 2)(x - 6)$

if $f(-2)=0$, what are all the factors of the function $f(x)=x^{3}-2x^{2}-68x - 120$? use the remainder theorem.\n$(x + 2)(x + 60)$\n$(x - 2)(x - 60)$\n$(x - 10)(x + 2)(x + 6)$\n$(x + 10)(x - 2)(x - 6)$

Answer

Explanation:

Step1: Apply Remainder Theorem

Since (f(-2)=0), by the Remainder Theorem, ((x + 2)) is a factor of (f(x)=x^{3}-2x^{2}-68x - 120).

Step2: Perform polynomial long - division

Divide (x^{3}-2x^{2}-68x - 120) by (x + 2). [ \begin{align*} \frac{x^{3}-2x^{2}-68x - 120}{x + 2}&=x^{2}-4x-60 \end{align*} ]

Step3: Factor the quadratic

Factor (x^{2}-4x - 60). We need two numbers that multiply to (-60) and add up to (-4). The numbers are (-10) and (6). So (x^{2}-4x - 60=(x - 10)(x+6)).

Answer:

((x - 10)(x + 2)(x + 6))