How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)2 have?(1 point) Responses 3 3 2 2 4 4 The degree cannot be determined from the information provided.

Answer 1

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

1. Zero from \( x \): The first factor \( x \) contributes one zero at \( x = 0 \).

2. Zero from \( (x-1) \): The second factor \( (x-1) \) contributes one zero at \( x = 1 \).

3. Zeros from \( (2x + 4)^2 \): The factor \( (2x + 4)^2 \) is a repeated zero. We first find the zero of \( 2x + 4 = 0 \): \[ 2x + 4 = 0 \implies 2x = -4 \implies x = -2 \] Since this factor is squared, it contributes two zeros at \( x = -2 \).

Now, combining all the zeros:

• From \( x \): 1 zero at \( x = 0 \)
• From \( (x-1) \): 1 zero at \( x = 1 \)
• From \( (2x + 4)^2 \): 2 zeros at \( x = -2 \)

Adding them up, we have: \[ 1 + 1 + 2 = 4 \]

Therefore, the total number of zeros, including repeated zeros, is 4.

The correct response is \( \boxed{4} \).