Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses

Question

the x-axis or touches the x-axis and turns around at each zero.

f(x)=5(x-2)(x+8)^2

The multiplicity of the smallest zero is

The multiplicity of the smallest zero is

and touches the x-axis and turns around at x=

Determine the zero(s).

The zero(s) is/are
(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 other zero is
(Simplify your answers.)
B. There are two zeros. The multiplicity of the largest zero is

(Simplify your answers.)
C. There is one zero. The multiplicity of the zero is
(Simplify your answer.)

Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the
answer boxes within your choice.

Answer 1

To find the zeros of the polynomial function \( f(x) = 5(x - 2)(x + 8)^2 \) and determine their multiplicities and behavior, follow these steps:

Step 1: Identify the Zeros

To find the zeros of the function, set \( f(x) = 0 \): \[ 5(x - 2)(x + 8)^2 = 0 \] This yields two factors to solve:

1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( (x + 8)^2 = 0 \) → \( x + 8 = 0 \) → \( x = -8 \)

Step 2: Find the Multiplicities

• The zero \( x = 2 \) comes from the factor \( (x - 2) \) which has a multiplicity of 1 (since it's linear).
• The zero \( x = -8 \) comes from the factor \( (x + 8)^2 \) which has a multiplicity of 2 (since it's a quadratic).

Summary of Zeros and Multiplicities

• Zeros: \( 2 \) and \( -8 \)
• Multiplicity of \( x = 2 \): 1
• Multiplicity of \( x = -8 \): 2

Determining the Behavior of the Graph at Each Zero

• For a zero with odd multiplicity, the graph crosses the x-axis.
• For a zero with even multiplicity, the graph touches the x-axis and turns around.

At the Zeros:

• At \( x = 2 \): The multiplicity is 1 (odd), so the graph crosses the x-axis.
• At \( x = -8 \): The multiplicity is 2 (even), so the graph touches the x-axis and turns around.

Final Answers

1. The zero(s) is/are:

   • \( -8, 2 \)

2. The multiplicity of the largest zero (\( x = 2 \)):

   • \( 1 \)

3. The multiplicity of the other zero (\( x = -8 \)):

   • \( 2 \)

4. The graph touches the x-axis and turns around at:

   • \( x = -8 \)

Selection Summary:

• Choose option A: There are two zeros. The multiplicity of the largest zero is 1, and the multiplicity of the other zero is 2.
• The function crosses the x-axis at \( x = 2 \) and touches the x-axis and turns around at \( x = -8 \).