Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :

Ask by Wade Lee. in the United States

Mar 20,2025

Question

Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :

Ask by Wade Lee. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

Factor the given function by first factoring out $$ x^3 $$:
$$
d(x) = x^5-4x^4+4x^3 = x^3\left(x^2-4x+4\right)
$$
Notice that the quadratic factors as:
$$
x^2-4x+4 = (x-2)^2
$$
Thus,
$$
d(x) = x^3 (x-2)^2
$$
The zeros of the function are the solutions to:
$$
x^3 = 0 \quad \text{and} \quad (x-2)^2 = 0
$$
This gives:
$$
x = 0 \quad \text{and} \quad x = 2
$$

Now, we state the multiplicities:
- The zero $$ x=0 $$ comes from the factor $$ x^3 $$ so it has multiplicity 3.
- The zero $$ x=2 $$ comes from the factor $$ (x-2)^2 $$ so it has multiplicity 2.

Thus, the zero(s) of $$ d $$ are:
$$
0 \text{ (multiplicity 3)}, \quad 2 \text{ (multiplicity 2)}
$$
The zeros of $$ d(x) $$ are $$ 0 $$ (with multiplicity 3) and $$ 2 $$ (with multiplicity 2).

Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :

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Find the zeros of the function and state the multiplicities. If there is more than one answer, separate them with commas. Select “None” if applicable. Part: Part 1 of 2 The zero(s) of :

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Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :