Questions: 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)=-3(x-7)(x+8)^3 Determine the zero(s). The zero(s) is/are 7, -8. (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.)

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)=-3(x-7)(x+8)^3

Determine the zero(s). The zero(s) is/are 7, -8. (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.)

Transcript text: 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. \[ f(x)=-3(x-7)(x+8)^{3} \] Determine the zero(s). The zero(s) is/are 7, -8 . (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 $\square$ - The multiplicity of the smallest zero is $\square$ . The multiplicity of the other zero is $\square$ (Simplify your answers.) B. There is one zero. The multiplicity of the zero is $\square$ . (Simplify your answer.) C. There are two zeros. The multiplicity of the largest zero is $\square$ . The multiplicity of the smallest zero is $\square$ . (Simplify your answers.)

Solution

Solution Steps

Step 1: Finding the Zeros

To find the zeros of the polynomial function $ f(x) = -3(x - 7)(x + 8)^{3} $, we set the function equal to zero:

\[ -3(x - 7)(x + 8)^{3} = 0 \]

This gives us two factors to consider:

$ x - 7 = 0 $ which leads to $ x = 7 $
$ (x + 8)^{3} = 0 $ which leads to $ x + 8 = 0 $ or $ x = -8 $

Thus, the zeros are $ x = -8 $ and $ x = 7 $.

Step 2: Determining the Multiplicities

Next, we determine the multiplicities of each zero based on the factors of the polynomial:

The zero $ x = 7 $ comes from the factor $ (x - 7) $, which has an exponent of 1. Therefore, the multiplicity is $ 1 $.
The zero $ x = -8 $ comes from the factor $ (x + 8)^{3} $, which has an exponent of 3. Therefore, the multiplicity is $ 3 $.

Step 3: Analyzing Graph Behavior

To analyze the behavior of the graph at each zero:

For $ x = 7 $ (multiplicity $ 1 $), since the multiplicity is odd, the graph crosses the x-axis.
For $ x = -8 $ (multiplicity $ 3 $), since the multiplicity is also odd, the graph crosses the x-axis.