Question

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

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

4
4

2
2

3

Answer 1

To determine how many zeros the function \( f(x) = x(x - 1)(2x + 4)^2 \) has, we can break it down into its factors.

1. The factor \( x \) provides 1 zero at \( x = 0 \).
2. The factor \( (x - 1) \) provides another zero at \( x = 1 \).
3. The factor \( (2x + 4)^2 \) is a squared term. The equation \( 2x + 4 = 0 \) gives us:
\[
2x = -4 \implies x = -2.
\]
Since this factor is squared, it contributes 2 zeros at \( x = -2 \).

Now, let's count the total number of zeros:

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

Adding these up gives:
\[
1 + 1 + 2 = 4 \text{ zeros}.
\]

Thus, the function \( f(x) = x(x - 1)(2x + 4)^2 \) has **4 zeros** in total, including repeated zeros.

The response is:
**4**.