Question

find the zeros of the polynomial function.

(f(x) = x^3 + 8x^2 - x - 8)

a. (x = 64)
b. (x = 1, x = -8, x = 8)
c. (x = -1, x = 1, x = -8)
d. (x = -8, x = 8)

Explanation:

Set the polynomial function to zero

To find the zeros, we set \(f(x) = 0\):

$$ x^3 + 8x^2 - x - 8 = 0 $$

Factor the polynomial by grouping

We group the terms to find common binomial factors:

$$ (x^3 + 8x^2) - (x + 8) = 0 $$

$$ x^2(x + 8) - 1(x + 8) = 0 $$

$$ (x^2 - 1)(x + 8) = 0 $$

Factor the difference of squares

We factor the quadratic term completely:

$$ (x - 1)(x + 1)(x + 8) = 0 $$

Solve for the zeros

Using the Finding Zeros of Factored Polynomials knowledge point:

$$
LATEXBLOCK0
$$

Answer:

• A. \(x = 64\)
• B. \(x = 1, x = -8, x = 8\)
• C. \(x = -1, x = 1, x = -8\) (Correct answer)
• D. \(x = -8, x = 8\)