QuestionFind the zeros and their multiplicities for the function $y=4 x^{3}(x+2)^{3}(x+1)$ .

Studdy Solution

STEP 1

Assumptions1. The function is $y=4x^{3}(x+)^{3}(x+1)$ . We need to find the zeros and their multiplicities

STEP 2

The zeros of a function are the x-values for which the function equals zero. We can find the zeros of the function by setting the function equal to zero and solving for x.
$4x^{}(x+2)^{}(x+1) =0$

STEP 3

A product of factors equals zero if and only if at least one of the factors equals zero. So, we can set each factor equal to zero and solve for x.
$x^{3} =0$ $(x+2)^{3} =0$ $(x+1) =0$

STEP 4

olving the first equation $x^{3} =0$ for x, we get $x =0$ The zero of the function is0 and its multiplicity is3 because the factor $x^{3}$ is raised to the power of3.

STEP 5

olving the second equation $(x+2)^{3} =0$ for x, we get $x = -2$ The zero of the function is -2 and its multiplicity is3 because the factor $(x+2)^{3}$ is raised to the power of3.

STEP 6

olving the third equation $(x+1) =0$ for x, we get $x = -1$ The zero of the function is -1 and its multiplicity is1 because the factor $(x+1)$ is raised to the power of1.
The zeros of the function $y=4x^{3}(x+2)^{3}(x+1)$ are0 with multiplicity3, -2 with multiplicity3, and -1 with multiplicity1.

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QuestionFind the zeros and their multiplicities for the function $y=4 x^{3}(x+2)^{3}(x+1)$ .

Studdy Solution

STEP 1

Assumptions1. The function is $y=4x^{3}(x+)^{3}(x+1)$ . We need to find the zeros and their multiplicities

STEP 2

The zeros of a function are the x-values for which the function equals zero. We can find the zeros of the function by setting the function equal to zero and solving for x.
$4x^{}(x+2)^{}(x+1) =0$

STEP 3

A product of factors equals zero if and only if at least one of the factors equals zero. So, we can set each factor equal to zero and solve for x.
$x^{3} =0$ $(x+2)^{3} =0$ $(x+1) =0$

STEP 4

olving the first equation $x^{3} =0$ for x, we get $x =0$ The zero of the function is0 and its multiplicity is3 because the factor $x^{3}$ is raised to the power of3.

STEP 5

olving the second equation $(x+2)^{3} =0$ for x, we get $x = -2$ The zero of the function is -2 and its multiplicity is3 because the factor $(x+2)^{3}$ is raised to the power of3.

STEP 6

olving the third equation $(x+1) =0$ for x, we get $x = -1$ The zero of the function is -1 and its multiplicity is1 because the factor $(x+1)$ is raised to the power of1.
The zeros of the function $y=4x^{3}(x+2)^{3}(x+1)$ are0 with multiplicity3, -2 with multiplicity3, and -1 with multiplicity1.

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QuestionFind the zeros and their multiplicities for the function y=4x3(x+2)3(x+1)y=4 x^{3}(x+2)^{3}(x+1)y=4x3(x+2)3(x+1).

Studdy Solution

STEP 1

Assumptions1. The function is y=4x3(x+)3(x+1)y=4x^{3}(x+)^{3}(x+1)y=4x3(x+)3(x+1). We need to find the zeros and their multiplicities

STEP 2

The zeros of a function are the x-values for which the function equals zero. We can find the zeros of the function by setting the function equal to zero and solving for x.4x(x+2)(x+1)=04x^{}(x+2)^{}(x+1) =04x(x+2)(x+1)=0

STEP 3

A product of factors equals zero if and only if at least one of the factors equals zero. So, we can set each factor equal to zero and solve for x.x3=0x^{3} =0x3=0(x+2)3=0(x+2)^{3} =0(x+2)3=0(x+1)=0(x+1) =0(x+1)=0

STEP 4

olving the first equation x3=0x^{3} =0x3=0 for x, we getx=0x =0x=0The zero of the function is0 and its multiplicity is3 because the factor x3x^{3}x3 is raised to the power of3.

STEP 5

olving the second equation (x+2)3=0(x+2)^{3} =0(x+2)3=0 for x, we getx=−2x = -2x=−2The zero of the function is -2 and its multiplicity is3 because the factor (x+2)3(x+2)^{3}(x+2)3 is raised to the power of3.

STEP 6

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QuestionFind the zeros and their multiplicities for the function $y=4 x^{3}(x+2)^{3}(x+1)$ .

Assumptions1. The function is $y=4x^{3}(x+)^{3}(x+1)$ . We need to find the zeros and their multiplicities

The zeros of a function are the x-values for which the function equals zero. We can find the zeros of the function by setting the function equal to zero and solving for x.
$4x^{}(x+2)^{}(x+1) =0$

A product of factors equals zero if and only if at least one of the factors equals zero. So, we can set each factor equal to zero and solve for x.
$x^{3} =0$ $(x+2)^{3} =0$ $(x+1) =0$

olving the first equation $x^{3} =0$ for x, we get $x =0$ The zero of the function is0 and its multiplicity is3 because the factor $x^{3}$ is raised to the power of3.

olving the second equation $(x+2)^{3} =0$ for x, we get $x = -2$ The zero of the function is -2 and its multiplicity is3 because the factor $(x+2)^{3}$ is raised to the power of3.