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.)
b. choose the correct graph of the function below. the scale for each graph is \\(-10, 10, 1\\) by \\(-80, 45, 5\\).

Explanation:

Find rational zeros

Using the Rational Root Theorem knowledge point

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

$$ f(1) = -(1)^3 - 2(1)^2 + 13(1) - 10 = -1 - 2 + 13 - 10 = 0 \implies x = 1 \text{ is a zero.} $$

Divide to find remaining factors

Using the Synthetic Division knowledge point

$$
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$$

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

Solve the quadratic factor

Using the Quadratic Formula knowledge point

$$ x^2 + 3x - 10 = 0 \implies (x + 5)(x - 2) = 0 \implies x = -5, \quad x = 2 $$

$$ \text{The complete set of zeros is } \{-5, 1, 2\}. $$

Analyze end behavior and intercepts

Using the End Behavior of Polynomials and Y-Intercept of Polynomial knowledge points

$$ \text{Leading term: } -x^3 \implies \lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = \infty $$

$$ y\text{-intercept: } f(0) = -10 $$

$$ x\text{-intercepts: } (-5, 0), (1, 0), (2, 0) $$

Match the correct graph

We look for a graph that:

1. Starts high on the left (\(x \to -\infty, y \to \infty\)) and ends low on the right (\(x \to \infty, y \to -\infty\)).
2. Crosses the \(x\)-axis at \(x = -5\), \(x = 1\), and \(x = 2\).
3. Has a \(y\)-intercept at \((0, -10)\).

• Graph A: Starts low on the left, ends high on the right (incorrect orientation).
• Graph B: Starts high on the left, ends low on the right. Crosses at \(x = -5\), \(x = 1\), and \(x = 2\).
• Graph C: Starts low on the left, ends low on the right (incorrect end behavior).
• Graph D: Starts low on the left, ends high on the right (incorrect orientation).

Therefore, Graph B is the correct representation.

Answer:

Question 1

The zeros are <blank>-5,1,2</blank>

Question 2

• (A) Graph starts low on the left, rises to the right
• (B) Graph starts high on the left, crosses at -5, 1, 2, and falls on the right (Correct answer)
• (C) Graph starts low on both ends
• (D) Graph starts low on the left, rises to the right with different intercepts