Deacon Hensley

2023-01-01

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

halliekorinnbsn

Expert

We're given that there are zeros at $x=-4$ (mult. 3) and at $x=0$ (mult. 1).
Since the zero at $x=-4$ has multiplicity three, we can write it as the factor $\left(x+4\right)$ raised to the third power:
$f\left(x\right)={\left(x+4\right)}^{3}$
$\underset{―}{f\left(x\right)=x{\left(x+4\right)}^{3}}$
Or, in expanded form, is
$\underset{―}{f\left(x\right)={x}^{4}+12{x}^{3}+48{x}^{2}+64x}$

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