# Write a polynomial f(x) that meets the given conditions. Polynomial of

Monique Slaughter 2021-12-13 Answered
Write a polynomial f(x) that meets the given conditions.
Polynomial of lowest degree with zeros of -1 (multiplicity 2) and -4 (multiplicity 2) and with f(0)=32
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Step 1
x=-1 is a zero so (x+1) is a factor. It has multiplicity = 2 , so ${\left(x+1\right)}^{2}$ is a factor of the polynomial f(x).
x=-4 is a zero so (x+4) is a factor. It has multiplicity = 2 , so ${\left(x+4\right)}^{2}$ is a factor of the polynomial f(x).
Step 2
Let $f\left(x\right)=k\cdot {\left(x+1\right)}^{2}\cdot {\left(x+4\right)}^{2}$
Given f(0)=32. Using this we find k.
$f\left(x\right)=k{\left(x+1\right)}^{2}{\left(x+4\right)}^{2}$
$f\left(0\right)=k{\left(0+1\right)}^{2}{\left(0+4\right)}^{2}$
$32=k{\left(1\right)}^{2}{\left(4\right)}^{2}$
32=16k
$k=\frac{32}{16}$
k=2
$f\left(x\right)=2{\left(x+1\right)}^{2}{\left(x+4\right)}^{2}$
Answer: $f\left(x\right)=2{\left(x+1\right)}^{2}{\left(x+4\right)}^{2}$.