Question

find the zeros for the polynomial function and give the multiplicity for each zero. state whether the graph crosses the x-axis or touches the x-axis and turns around at each zero.

(f(x) = 6(x - 9)(x + 2)^3)

determine the zero(s).

the zero(s) is/are 9, -2
(type integers or decimals. use a comma to separate answers as needed.)

determine the multiplicities of the zero(s). select the correct choice below and, if necessary, fill in the answer box(es) within your choice.

a. there are three zeros. the multiplicity of the largest zero is . the multiplicity of the smallest zero is . the multiplicity of the other zero is .
(simplify your answers.)

b. there is one zero. the multiplicity of the zero is .
(simplify your answer.)

c. there are two zeros. the multiplicity of the largest zero is . the multiplicity of the smallest zero is .
(simplify your answers.)

Explanation:

Find the zeros of the function

Using the Polynomial Functions knowledge point

$$ f(x) = 6(x-9)(x+2)^3 $$

Setting \(f(x) = 0\):

$$ 6(x-9)(x+2)^3 = 0 \implies x = 9 \quad \text{or} \quad x = -2 $$

Determine the multiplicities

Using the Multiplicity of Zeros knowledge point

$$
LATEXBLOCK0
$$

The largest zero is \(9\) with multiplicity \(1\).
The smallest zero is \(-2\) with multiplicity \(3\).
This matches choice C.

Determine the graph behavior at each zero

Using the Graph Behavior at Zeros knowledge point

$$
LATEXBLOCK1
$$

Thus, the graph crosses the x-axis at both zeros.

Answer:

Question 1

The zero(s) is/are <blank>\(9, -2\)</blank>.

Question 2

• A. There are three zeros. The multiplicity of the largest zero is <blank></blank>. The multiplicity of the smallest zero is <blank></blank>. The multiplicity of the other zero is <blank></blank>.
• B. There is one zero. The multiplicity of the zero is <blank></blank>.
• C. There are two zeros. The multiplicity of the largest zero is <blank>\(1\)</blank>. The multiplicity of the smallest zero is <blank>\(3\)</blank>. (Correct answer)

Question 3

• A. The graph crosses the x-axis at each zero. (Correct answer)
• B. The graph touches the x-axis and turns around at each zero.
• C. The graph crosses the x-axis at \(x = 9\) and touches the x-axis and turns around at \(x = -2\).
• D. The graph crosses the x-axis at \(x = -2\) and touches the x-axis and turns around at \(x = 9\).