Question

How do you find a polynomial function of degree $4$ with $-4$ as a zero of multiplicity $3$ and $0$ as a zero of multiplicity $1$?

Answer 1

Polynomial function:

Given zeros are $x=-4$(multiply $3$) and at $x=0$(multiply $1$)

Since the zeros $x=-4$ have a multiplicity of three, we can write it as the factor $(x+4)$ raised to the third power.

$f\left(x\right)={(x+4)}^3$

And there is also zero at $x=0$, so we can add an $x$.

$f\left(x\right)=x{(x+4)}^3$

In expanded form, is

$f\left(x\right)=x^4+12x^3+48x^2+64x$

Hence, the polynomial function of degree $4$ with $-4$ as a zero of multiplicity $3$ and $0$ as a zero of multiplicity $1$ is

$f\left(x\right)=x^4+12x^3+48x^2+64x$