Question

an incomplete graph of the polynomial function \\(f(x) = -x^3 - 2x^2 + 13x - 10\\) is shown on the right.
a. find all zeros of the function.
b. without using a graphing utility, draw a complete graph of the function.

a. the zeros are
(use a comma to separate answers as needed.)

Explanation:

Identify potential rational zeros

Using the Rational Root Theorem knowledge point

$$ \text{Possible rational roots of } f(x) = -x^3 - 2x^2 + 13x - 10 \text{ are factors of } -10: \pm 1, \pm 2, \pm 5, \pm 10. $$

Test potential roots to find a zero

Using the Rational Root Theorem knowledge point

$$ f(1) = -(1)^3 - 2(1)^2 + 13(1) - 10 = -1 - 2 + 13 - 10 = 0 $$

Perform polynomial division

Using the Synthetic Division knowledge point

$$
LATEXBLOCK0
$$

$$ f(x) = (x - 1)(-x^2 - 3x + 10) = -(x - 1)(x^2 + 3x - 10) $$

Factor the remaining quadratic expression

Using the Factoring Quadratic Trinomials knowledge point

$$ x^2 + 3x - 10 = (x + 5)(x - 2) $$

Determine all zeros of the function

Using the Finding Zeros of Factored Polynomials knowledge point

$$ f(x) = -(x - 1)(x + 5)(x - 2) = 0 \implies x = 1, -5, 2 $$

Answer:

a. The zeros are <blank>-5, 1, 2</blank>
(Use a comma to separate answers as needed.)