Question

How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)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 find how many zeros the function \( f(x) = x(x - 1)(2x + 4)^2 \) has, we need to analyze each factor of the function for its zeros.

1. **The factor \( x \)**:
- This factor gives us a zero at \( x = 0 \) (with a multiplicity of 1).

2. **The factor \( x - 1 \)**:
- This factor gives us a zero at \( x = 1 \) (with a multiplicity of 1).

3. **The factor \( (2x + 4)^2 \)**:
- To find the zeros, we set \( 2x + 4 = 0 \).
- Solving this gives \( 2x = -4 \) or \( x = -2 \).
- The square indicates that this zero has a multiplicity of 2.

Now we can count the total number of zeros:

- From \( x \) : 1 zero (at \( x = 0 \))
- From \( x - 1 \): 1 zero (at \( x = 1 \))
- From \( (2x + 4)^2 \): 2 zeros (at \( x = -2 \), counted twice due to the square)

So the total number of zeros, including repeated zeros, is:
1 (from \( x \)) + 1 (from \( x - 1 \)) + 2 (from \( (2x + 4)^2 \)) = **4 zeros**.

Thus, the correct response is:

**4**.