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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)=4(x-6)(x+7)^3
Determine the​ zero(s).
The​ zero(s) is/are enter your response here
.
​(Type integers or decimals. Use a comma to separate answers as​ needed.)
Part 2
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 two zeros. The multiplicity of the largest

zero is enter your response here
.
The multiplicity of the smallest

zero is enter your response here
.
​(Simplify your​ answers.)
B.
There is one zero. The multiplicity of the zero is enter your response here
.
​(Simplify your​ answer.)
C.
There are three zeros. The multiplicity of the largest

zero is enter your response here
.
The multiplicity of the smallest

zero is enter your response here
.
The multiplicity of the other zero is enter your response here
​(Simplify your​ answers.)

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.
A.
The graph crosses the​ x-axis at xequals
enter your response here
and touches the​ x-axis and turns around at xequals
enter your response here
.
​(Simplify your answers. Type integers or decimals. Use a comma to separate answers as​ needed.)
B.
The graph touches the​ x-axis and turns around at all zeros.
C.
The graph crosses the​ x-axis at all zeros.

1 answer

To find the zeros and their multiplicities for the polynomial function \( f(x) = 4(x - 6)(x + 7)^3 \), let's start by identifying the zeros.

Part 1: Zeros

  1. Set the function equal to zero: \[ f(x) = 4(x - 6)(x + 7)^3 = 0 \] This gives us two factors to consider:
    • The first factor \( (x - 6) \) gives a zero at \( x = 6 \).
    • The second factor \( (x + 7)^3 \) gives a zero at \( x = -7 \).

Thus, the zeros are: \[ 6, -7 \]

Part 2: Multiplicities

Now we can determine the multiplicities:

  • The zero \( x = 6 \) comes from the factor \( (x - 6) \), which has a multiplicity of 1.
  • The zero \( x = -7 \) comes from the factor \( (x + 7)^3 \), which has a multiplicity of 3.

Summary of Multiplicities:

  • The multiplicity of the zero at \( x = 6 \) is \( 1 \) (smallest zero).
  • The multiplicity of the zero at \( x = -7 \) is \( 3 \) (largest zero).

Behavior of the Function at Each Zero

The behavior of the graph at each zero is determined by the multiplicity:

  • A zero with an odd multiplicity will cross the x-axis.
  • A zero with an even multiplicity will touch the x-axis and turn around.

Thus:

  • At \( x = 6 \) (multiplicity is 1, which is odd), the graph crosses the x-axis.
  • At \( x = -7 \) (multiplicity is 3, which is odd), the graph also crosses the x-axis.

Final Answers

  1. Zeros: \[ 6, -7 \]

  2. Multiplicities:

    • There are two zeros. The multiplicity of the largest zero (6) is \( 1 \). The multiplicity of the smallest zero (-7) is \( 3 \).

  3. Behavior:

    • The graph crosses the x-axis at \( x = 6 \) and crosses the x-axis at \( x = -7 \).

Putting it all together, your answer will be:

  • Zeros: \( 6, -7 \)
  • Multiplicities:
    • The multiplicity of the largest zero is \( 1 \).
    • The multiplicity of the smallest zero is \( 3 \).
  • Behavior:
    • The graph crosses the x-axis at \( x = 6 \) and \( x = -7 \).