# Find a polynomial of the specified degree that has the given zeros. Degree 4, zeros -2, 0, 2, 4

Find a polynomial of the specified degree that has the given zeros. Degree 4, zeros -2, 0, 2, 4
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Recall, according to the factor theorem the expression $\left(x–c\right)$ is a factor of a polynomial P(x) if and only if P(c) = 0.
1) Let a polynomial function P(x):
If c = -2 is a zero of P(x) the expression $\left(x+2\right)$ is a factor of P(x).
If c = 0 is a zero of P(x) the expression $\left(x–0\right)=$ is a factor of P(x).
If c = 2 is a zero of P(x) the expression $\left(x–2\right)=$ is a factor of P(x).
If c = 4 is a zero of P(x) the expression $\left(x–4\right)=$ is a factor of P(x).
2) The polynomial P(x) can be written in factored form as:
P(x) $=x\left(x–4\right)\left(x+2\right)\left(x–2\right)$
P(x) $=\left({x}^{2}–4x\right)\left(x+2\right)\left(x–2\right)$
3) Apply the difference of squares identity: ${a}^{2}–{b}^{2}=\left(a+b\right)\left(a–b\right)$
P(x) $=\left({x}^{2}–4x\right)\left({x}^{2}–4\right)$
P(x) $={x}^{4}–4{x}^{3}–4{x}^{2}+16x$