Question

Given the zeros of polynomial $P$, then list their multiplicities:
$P\left(x\right)=-x{(-x-2)}^2(x+2)$

Answer 1

A polynomial is a function of the form $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+...+a_2{x}^2+a_{1}x+a_0$

where $a_n,a_{n-1},...a_2,a_1,a_0$ are constants and $n$ is a natural number.

To get Zeroes of the polynomial take $f\left(x\right)=0$

Now given that

$P\left(x\right)=-x(-x-2{)}^2(x+2)$

or $P\left(x\right)=-x(x+2{)}^2(x+2)$

Zeroes of $P\left(x\right)$ are $0,-2,-2,-2$

The zero $-2$ has multiplicity $3$, and

Zero $0$ has multiplicity $1$.