Question

How many zeros does the function f(x)=x(x−1)(2x+4)2 have?(1 point)ResponsesThe degree cannot be determined from the information provided.The degree cannot be determined from the information provided.44332

Answer 1

To determine how many zeros the function \( f(x) = x(x - 1)(2x + 4)^2 \) has, we can analyze each factor in the function separately.

1. **First factor**: \( x \)
This factor has a root at \( x = 0 \).

2. **Second factor**: \( (x - 1) \)
This factor has a root at \( x = 1 \).

3. **Third factor**: \( (2x + 4)^2 \)
To find the roots for this factor, set it to zero:
\[
2x + 4 = 0 \implies 2x = -4 \implies x = -2
\]
Since this factor is squared, the root \( x = -2 \) counts as two zeros.

Now we can tally the total number of zeros from each of the factors:

- From \( x \): 1 root (at \( x = 0 \))
- From \( x - 1 \): 1 root (at \( x = 1 \))
- From \( (2x + 4)^2 \): 2 roots (at \( x = -2 \))

Adding these together gives:
\[
1 + 1 + 2 = 4
\]

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

Therefore, the correct response is **4**.