Math / Algebra

QuestionUse the given polynomial function to identify the zeros of the function and the multiplicity of each zero Leave any remaining answer boxes empty. $f(x)=-4 x^{2}(x+6)^{2}(x-3)^{2}$ \begin{tabular}{|c|c|} \hline Zeros & Mult. \\ \hline $\square$ & $\square$ \\ $\square$ & $\square$ \\ $\square$ & $\square$ \\ $\square$ & $\square$ \\ $\square$ & \\ $\square$ \\ \hline \end{tabular}
Note: It is possible that some of the answer boxes will be empty!

Studdy Solution

STEP 1

1. The polynomial function is given in factored form.
2. The zeros of the function are the values of $x$ that make the function equal to zero.
3. The multiplicity of a zero is the power of the factor associated with that zero.

STEP 2

1. Identify the factors of the polynomial.
2. Determine the zeros from the factors.
3. Determine the multiplicity of each zero.
4. Fill in the answer boxes with the zeros and their multiplicities.

STEP 3

Identify the factors of the polynomial function $f(x) = -4x^2(x+6)^2(x-3)^2$ .
The factors are: - $x^2$ - $(x+6)^2$ - $(x-3)^2$

STEP 4

Determine the zeros from the factors:
- The factor $x^2$ gives the zero $x = 0$ . - The factor $(x+6)^2$ gives the zero $x = -6$ . - The factor $(x-3)^2$ gives the zero $x = 3$ .

STEP 5

Determine the multiplicity of each zero:
- The zero $x = 0$ has multiplicity 2 (from $x^2$ ). - The zero $x = -6$ has multiplicity 2 (from $(x+6)^2$ ). - The zero $x = 3$ has multiplicity 2 (from $(x-3)^2$ ).

STEP 6

Fill in the answer boxes with the zeros and their multiplicities:
\begin{tabular}{|c|c|} \hline Zeros & Mult. \\ \hline 0 & 2 \\ -6 & 2 \\ 3 & 2 \\ \square & \square \\ \square & \\ \square & \\ \hline \end{tabular}
The zeros of the function are $x = 0, -6,$ and $3$ with multiplicities of 2 each.

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Studdy Solution

STEP 1

1. The polynomial function is given in factored form.
2. The zeros of the function are the values of $x$ that make the function equal to zero.
3. The multiplicity of a zero is the power of the factor associated with that zero.

STEP 2

1. Identify the factors of the polynomial.
2. Determine the zeros from the factors.
3. Determine the multiplicity of each zero.
4. Fill in the answer boxes with the zeros and their multiplicities.

STEP 3

Identify the factors of the polynomial function $f(x) = -4x^2(x+6)^2(x-3)^2$ .
The factors are: - $x^2$ - $(x+6)^2$ - $(x-3)^2$

STEP 4

Determine the zeros from the factors:
- The factor $x^2$ gives the zero $x = 0$ . - The factor $(x+6)^2$ gives the zero $x = -6$ . - The factor $(x-3)^2$ gives the zero $x = 3$ .

STEP 5

Determine the multiplicity of each zero:
- The zero $x = 0$ has multiplicity 2 (from $x^2$ ). - The zero $x = -6$ has multiplicity 2 (from $(x+6)^2$ ). - The zero $x = 3$ has multiplicity 2 (from $(x-3)^2$ ).

STEP 6

Fill in the answer boxes with the zeros and their multiplicities:
\begin{tabular}{|c|c|} \hline Zeros & Mult. \\ \hline 0 & 2 \\ -6 & 2 \\ 3 & 2 \\ \square & \square \\ \square & \\ \square & \\ \hline \end{tabular}
The zeros of the function are $x = 0, -6,$ and $3$ with multiplicities of 2 each.

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Studdy Solution

STEP 1

1. The polynomial function is given in factored form.2. The zeros of the function are the values of x x x that make the function equal to zero.3. The multiplicity of a zero is the power of the factor associated with that zero.

STEP 2

1. Identify the factors of the polynomial.2. Determine the zeros from the factors.3. Determine the multiplicity of each zero.4. Fill in the answer boxes with the zeros and their multiplicities.

STEP 3

Identify the factors of the polynomial function f(x)=−4x2(x+6)2(x−3)2 f(x) = -4x^2(x+6)^2(x-3)^2 f(x)=−4x2(x+6)2(x−3)2.The factors are: - x2 x^2 x2 - (x+6)2 (x+6)^2 (x+6)2 - (x−3)2 (x-3)^2 (x−3)2

STEP 4

Determine the zeros from the factors:- The factor x2 x^2 x2 gives the zero x=0 x = 0 x=0. - The factor (x+6)2 (x+6)^2 (x+6)2 gives the zero x=−6 x = -6 x=−6. - The factor (x−3)2 (x-3)^2 (x−3)2 gives the zero x=3 x = 3 x=3.

STEP 5

Determine the multiplicity of each zero:- The zero x=0 x = 0 x=0 has multiplicity 2 (from x2 x^2 x2). - The zero x=−6 x = -6 x=−6 has multiplicity 2 (from (x+6)2 (x+6)^2 (x+6)2). - The zero x=3 x = 3 x=3 has multiplicity 2 (from (x−3)2 (x-3)^2 (x−3)2).

STEP 6

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1. The polynomial function is given in factored form.
2. The zeros of the function are the values of $x$ that make the function equal to zero.
3. The multiplicity of a zero is the power of the factor associated with that zero.

1. Identify the factors of the polynomial.
2. Determine the zeros from the factors.
3. Determine the multiplicity of each zero.
4. Fill in the answer boxes with the zeros and their multiplicities.

Identify the factors of the polynomial function $f(x) = -4x^2(x+6)^2(x-3)^2$ .
The factors are: - $x^2$ - $(x+6)^2$ - $(x-3)^2$

Determine the zeros from the factors:
- The factor $x^2$ gives the zero $x = 0$ . - The factor $(x+6)^2$ gives the zero $x = -6$ . - The factor $(x-3)^2$ gives the zero $x = 3$ .

Determine the multiplicity of each zero:
- The zero $x = 0$ has multiplicity 2 (from $x^2$ ). - The zero $x = -6$ has multiplicity 2 (from $(x+6)^2$ ). - The zero $x = 3$ has multiplicity 2 (from $(x-3)^2$ ).