Question

Given the zeros of polynomial $P$ and list their multiplicities:
$P\left(x\right)={(x+5)}^3(x-6)(-x+6)$

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

$P\left(x\right)=(x+5{)}^3(x-6\left)\right(-x+6)$

or $P\left(x\right)=-(x+5{)}^3(x-6{)}^2$

Zeroes of $P\left(x\right)$ are $-5,-5,-5,6,6$

The zero $-5$ has multiplicity $3$, and

Zero $6$ has multiplicity $2$.