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.

2
2

3
3

4

Answer 1

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

1. The first factor is \( x \). This contributes one zero at \( x = 0 \).
2. The second factor is \( (x - 1) \). This contributes one zero at \( x = 1 \).
3. The third factor is \( (2x + 4)^2 \). The term \( (2x + 4) \) can be set to zero to find its zeros: \[ 2x + 4 = 0 \implies 2x = -4 \implies x = -2. \] Since this factor is squared, it contributes two zeros (including repeated zeros) at \( x = -2 \).

Now, let's summarize the zeros:

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

Adding these together gives:

• Total zeros = \( 1 + 1 + 2 = 4 \).

Thus, the answer is 4.