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Math  /  Algebra

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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)=3(x−6)(x+1)3f(x)=3(x-6)(x+1)^{3}
Determine the zero(s). The zero(s) is/are □\square (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 two zeros. The multiplicity of the largest zero is □\square . The multiplicity of the smallest zero is □\square . (Simplify your answers.) B. 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.) C. There is one zero. The multiplicity of the zero is □\square . (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. A. The graph touches the xx-axis and turns around at all zeros. B. The graph crosses the xx-axis at all zeros. C. The graph crosses the xx-axis at x=x= □\square and touches the xx-axis and turns around at x=x= □\square I. (Simplify your answers. Type integers or decimals. Use a comma to separate answers as needed.)

Studdy Solution

STEP 1

1. The function given is a polynomial f(x)=3(x−6)(x+1)3 f(x) = 3(x-6)(x+1)^3 .
2. Zeros of the polynomial occur where the function equals zero.
3. The multiplicity of a zero is the exponent of the factor in the polynomial.
4. The behavior of the graph at each zero depends on the multiplicity.

STEP 2

1. Find the zeros of the polynomial.
2. Determine the multiplicity of each zero.
3. Determine the behavior of the graph at each zero.

STEP 3

To find the zeros of the polynomial f(x)=3(x−6)(x+1)3 f(x) = 3(x-6)(x+1)^3 , set f(x)=0 f(x) = 0 :
3(x−6)(x+1)3=0 3(x-6)(x+1)^3 = 0
This equation is satisfied when any of the factors is zero.

STEP 4

Solve each factor for zero:
1. x−6=0 x-6 = 0 gives x=6 x = 6 .
2. (x+1)3=0 (x+1)^3 = 0 gives x=−1 x = -1 .
The zeros of the polynomial are x=6 x = 6 and x=−1 x = -1 .

STEP 5

Determine the multiplicity of each zero:
1. The zero x=6 x = 6 comes from the factor (x−6) (x-6) , which has an implicit exponent of 1. Thus, the multiplicity is 1.
2. The zero x=−1 x = -1 comes from the factor (x+1)3 (x+1)^3 , which has an exponent of 3. Thus, the multiplicity is 3.

STEP 6

Determine the behavior of the graph at each zero:
1. If the multiplicity is odd, the graph crosses the x-axis at that zero.
2. If the multiplicity is even, the graph touches the x-axis and turns around at that zero.
- For x=6 x = 6 with multiplicity 1 (odd), the graph crosses the x-axis. - For x=−1 x = -1 with multiplicity 3 (odd), the graph crosses the x-axis.
The zero(s) is/are 6,−1 \boxed{6, -1} .
There are two zeros. The multiplicity of the largest zero is 1 \boxed{1} . The multiplicity of the smallest zero is 3 \boxed{3} .
The graph crosses the x x -axis at x=6 x = \boxed{6} and crosses the x x -axis at x=−1 x = \boxed{-1} .

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