Question

if \\(f(-2) = 0\\), what are all the factors of the function \\(f(x) = x^3 - 2x^2 - 68x - 120\\)? use the remainder theorem.

\\((x + 2)(x + 60)\\)
\\((x - 2)(x - 60)\\)
\\((x - 10)(x + 2)(x + 6)\\)
\\((x + 10)(x - 2)(x - 6)\\)

Explanation:

Identify the first factor

Using the Factor Theorem knowledge point

$$ f(-2) = 0 \implies (x - (-2)) = (x + 2) \text{ is a factor of } f(x) $$

Divide the polynomial

Using the Polynomial Factoring knowledge point

$$
LATEXBLOCK0
$$

Factor the remaining quadratic

Using the Polynomial Factoring knowledge point

$$
LATEXBLOCK1
$$

Combine all factors

Using the Polynomial Factoring knowledge point

$$ f(x) = (x - 10)(x + 2)(x + 6) $$

Answer:

• (A) \((x + 2)(x + 60)\)
• (B) \((x - 2)(x - 60)\)
• (C) \((x - 10)(x + 2)(x + 6)\) (Correct answer)
• (D) \((x + 10)(x - 2)(x - 6)\)