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

Question

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.