Question

How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)2 have?(1 point) Responses 3 3 2 2 4 4 The degree cannot be determined from the information provided.

Answer 1

To find the number of zeros, including repeated zeros, for the function \( f(x) = x(x-1)(2x+4)^2 \), we need to analyze each factor of the function.

1. **Zero from \( x \)**: The first factor \( x \) contributes one zero at \( x = 0 \).

2. **Zero from \( (x-1) \)**: The second factor \( (x-1) \) contributes one zero at \( x = 1 \).

3. **Zeros from \( (2x + 4)^2 \)**: The factor \( (2x + 4)^2 \) is a repeated zero. We first find the zero of \( 2x + 4 = 0 \):
\[
2x + 4 = 0 \implies 2x = -4 \implies x = -2
\]
Since this factor is squared, it contributes two zeros at \( x = -2 \).

Now, combining all the zeros:

- From \( x \): 1 zero at \( x = 0 \)
- From \( (x-1) \): 1 zero at \( x = 1 \)
- From \( (2x + 4)^2 \): 2 zeros at \( x = -2 \)

Adding them up, we have:
\[
1 + 1 + 2 = 4
\]

Therefore, the total number of zeros, including repeated zeros, is **4**.

The correct response is \( \boxed{4} \).