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

Answer 1

To determine the number of zeros of a function, we need to examine the factors of the function.

The function $f(x)$ has zeros when any of its factors equal zero.

The first factor, $x$, equals zero when $x = 0$. So, $f(x)$ has one zero at $x=0$.

The second factor, $(x-1)$, equals zero when $x=1$. So, $f(x)$ has one zero at $x=1$.

The third factor, $(2x+4)$, equals zero when $x=-2$. So, $f(x)$ has one zero at $x=-2$.

Since $(2x+4)$ is squared, $f(x)$ will have an additional zero at $x=-2$ (a repeated zero).

Therefore, $f(x)$ has a total of $\boxed{4}$ zeros, including repeated zeros.