Question

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

2
2

4
4

3
3

The degree cannot be determined from the information provided.

Answer 1

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

1. **The first factor** is \( x \). This factor has a zero at \( x = 0 \).

2. **The second factor** is \( (x - 1) \). This factor has a zero at \( x = 1 \).

3. **The third factor** is \( (2x + 4)^2 \). The linear factor \( 2x + 4 = 0 \) gives a zero at:
\[
2x + 4 = 0 \implies 2x = -4 \implies x = -2
\]
Since this factor is squared, the zero at \( x = -2 \) has a multiplicity of 2.

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

- 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)

Adding these together, we get:
\[
1 + 1 + 2 = 4
\]

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

The correct answer is **4**.