Math / Algebra

QuestionFind the zeros of $f(x)=(x-2)^{5}(x+5)^{3}$ and state the multiplicity. 2 , multiplicity $5 ;-5$ , multiplicity 3 5 , multiplicity $2 ;-5$ , multiplicity 3 5 , multiplicity $2 ; 3$ , multiplicity -5 2 , multiplicity $5 ; 3$ , multiplicity -5

Studdy Solution

STEP 1

1. The function given is $f(x) = (x-2)^5(x+5)^3$ .
2. We need to find the zeros of this function.
3. We also need to determine the multiplicity of each zero.

STEP 2

1. Identify the factors of the function.
2. Determine the zeros of the function.
3. Determine the multiplicity of each zero.
4. Match the zeros and their multiplicities with the given options.

STEP 3

Identify the factors of the function $f(x) = (x-2)^5(x+5)^3$ .
The factors are $(x-2)$ and $(x+5)$ .

STEP 4

Determine the zeros of the function by setting each factor equal to zero.
For $(x-2) = 0$ , the zero is $x = 2$ .
For $(x+5) = 0$ , the zero is $x = -5$ .

STEP 5

Determine the multiplicity of each zero.
The zero $x = 2$ comes from the factor $(x-2)^5$ , so its multiplicity is $5$ .
The zero $x = -5$ comes from the factor $(x+5)^3$ , so its multiplicity is $3$ .

STEP 6

Match the zeros and their multiplicities with the given options.
The correct option is: 2, multiplicity 5; -5, multiplicity 3
The zeros of the function are $x = 2$ with multiplicity 5, and $x = -5$ with multiplicity 3.

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QuestionFind the zeros of $f(x)=(x-2)^{5}(x+5)^{3}$ and state the multiplicity. 2 , multiplicity $5 ;-5$ , multiplicity 3 5 , multiplicity $2 ;-5$ , multiplicity 3 5 , multiplicity $2 ; 3$ , multiplicity -5 2 , multiplicity $5 ; 3$ , multiplicity -5

Studdy Solution

STEP 1

1. The function given is $f(x) = (x-2)^5(x+5)^3$ .
2. We need to find the zeros of this function.
3. We also need to determine the multiplicity of each zero.

STEP 2

1. Identify the factors of the function.
2. Determine the zeros of the function.
3. Determine the multiplicity of each zero.
4. Match the zeros and their multiplicities with the given options.

STEP 3

Identify the factors of the function $f(x) = (x-2)^5(x+5)^3$ .
The factors are $(x-2)$ and $(x+5)$ .

STEP 4

Determine the zeros of the function by setting each factor equal to zero.
For $(x-2) = 0$ , the zero is $x = 2$ .
For $(x+5) = 0$ , the zero is $x = -5$ .

STEP 5

Determine the multiplicity of each zero.
The zero $x = 2$ comes from the factor $(x-2)^5$ , so its multiplicity is $5$ .
The zero $x = -5$ comes from the factor $(x+5)^3$ , so its multiplicity is $3$ .

STEP 6

Match the zeros and their multiplicities with the given options.
The correct option is: 2, multiplicity 5; -5, multiplicity 3
The zeros of the function are $x = 2$ with multiplicity 5, and $x = -5$ with multiplicity 3.

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Studdy Solution

STEP 1

1. The function given is f(x)=(x−2)5(x+5)3 f(x) = (x-2)^5(x+5)^3 f(x)=(x−2)5(x+5)3.2. We need to find the zeros of this function.3. We also need to determine the multiplicity of each zero.

STEP 2

1. Identify the factors of the function.2. Determine the zeros of the function.3. Determine the multiplicity of each zero.4. Match the zeros and their multiplicities with the given options.

STEP 3

Identify the factors of the function f(x)=(x−2)5(x+5)3 f(x) = (x-2)^5(x+5)^3 f(x)=(x−2)5(x+5)3.The factors are (x−2) (x-2) (x−2) and (x+5) (x+5) (x+5).

STEP 4

Determine the zeros of the function by setting each factor equal to zero.For (x−2)=0 (x-2) = 0 (x−2)=0, the zero is x=2 x = 2 x=2.For (x+5)=0 (x+5) = 0 (x+5)=0, the zero is x=−5 x = -5 x=−5.

STEP 5

Determine the multiplicity of each zero.The zero x=2 x = 2 x=2 comes from the factor (x−2)5 (x-2)^5 (x−2)5, so its multiplicity is 5 5 5.The zero x=−5 x = -5 x=−5 comes from the factor (x+5)3 (x+5)^3 (x+5)3, so its multiplicity is 3 3 3.

STEP 6

Match the zeros and their multiplicities with the given options.The correct option is: 2, multiplicity 5; -5, multiplicity 3 The zeros of the function are x=2 x = 2 x=2 with multiplicity 5, and x=−5 x = -5 x=−5 with multiplicity 3.

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1. The function given is $f(x) = (x-2)^5(x+5)^3$ .
2. We need to find the zeros of this function.
3. We also need to determine the multiplicity of each zero.

1. Identify the factors of the function.
2. Determine the zeros of the function.
3. Determine the multiplicity of each zero.
4. Match the zeros and their multiplicities with the given options.

Identify the factors of the function $f(x) = (x-2)^5(x+5)^3$ .
The factors are $(x-2)$ and $(x+5)$ .

Determine the zeros of the function by setting each factor equal to zero.
For $(x-2) = 0$ , the zero is $x = 2$ .
For $(x+5) = 0$ , the zero is $x = -5$ .

Determine the multiplicity of each zero.
The zero $x = 2$ comes from the factor $(x-2)^5$ , so its multiplicity is $5$ .
The zero $x = -5$ comes from the factor $(x+5)^3$ , so its multiplicity is $3$ .

Match the zeros and their multiplicities with the given options.
The correct option is: 2, multiplicity 5; -5, multiplicity 3
The zeros of the function are $x = 2$ with multiplicity 5, and $x = -5$ with multiplicity 3.