Question

How many zeros, including repeated zeros, does the f\left(x\right)=-x\left(x+2\right)^2

Answer 1

To find the zeros of the function \( f(x) = -x(x+2)^2 \), we need to determine the values of \( x \) that make \( f(x) = 0 \).

The function is a product of two factors: \( -x \) and \( (x+2)^2 \).

1. Set the first factor equal to zero:
\[
-x = 0 \implies x = 0
\]

2. Set the second factor equal to zero:
\[
(x+2)^2 = 0 \implies x + 2 = 0 \implies x = -2
\]

Now we determine the multiplicity of the roots.

- The root \( x = 0 \) occurs once (multiplicity 1).
- The root \( x = -2 \) occurs twice because it is a square (multiplicity 2).

Now, we count the total number of zeros including their multiplicities:
- \( x = 0 \) contributes 1 zero.
- \( x = -2 \) contributes 2 zeros.

Thus, the total number of zeros, including repeated zeros, is:
\[
1 + 2 = 3
\]

Therefore, the answer is \( \boxed{3} \).