Question

How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)2
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2
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2
have?(1 point)
Responses

4
4

3
3

The degree cannot be determined from the information provided.
The degree cannot be determined from the information provided.

2

Answer 1

To determine how many zeros the function \( f(x) = x(x - 1)(2x + 4)^2 \) has, we need to find the roots of the function.

1. **Finding the roots:**
- The first term \( x \) gives a root at \( x = 0 \).
- The second term \( x - 1 \) gives a root at \( x = 1 \).
- The third term \( (2x + 4)^2 \) gives a root when \( 2x + 4 = 0 \):
\[
2x + 4 = 0 \implies 2x = -4 \implies x = -2
\]
Since this term is squared, it indicates that the root \( x = -2 \) has a multiplicity of 2.

2. **Counting the zeros:**
- The root \( x = 0\) contributes 1 zero.
- The root \( x = 1\) contributes 1 zero.
- The root \( x = -2\) contributes 2 zeros due to its multiplicity.

Adding these together:
- From \( x = 0\): 1 zero
- From \( x = 1\): 1 zero
- From \( x = -2\): 2 zeros

Therefore, the total count of zeros (including repeats) is:
\[
1 + 1 + 2 = 4
\]

Thus, the answer is **4**.