Question

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

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

2021-02-26
Recall, according to the factor theorem the expression $$\displaystyle{\left({x}–{c}\right)}$$ 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 $$\displaystyle{\left({x}+{2}\right)}$$ is a factor of P(x).
If c = 0 is a zero of P(x) the expression $$\displaystyle{\left({x}–{0}\right)}=$$ is a factor of P(x).
If c = 2 is a zero of P(x) the expression $$\displaystyle{\left({x}–{2}\right)}=$$ is a factor of P(x).
If c = 4 is a zero of P(x) the expression $$\displaystyle{\left({x}–{4}\right)}=$$ is a factor of P(x).
2) The polynomial P(x) can be written in factored form as:
P(x) $$\displaystyle={x}{\left({x}–{4}\right)}{\left({x}+{2}\right)}{\left({x}–{2}\right)}$$
P(x) $$\displaystyle={\left({x}^{{2}}–{4}{x}\right)}{\left({x}+{2}\right)}{\left({x}–{2}\right)}$$
3) Apply the difference of squares identity: $$\displaystyle{a}^{{2}}–{b}^{{2}}={\left({a}+{b}\right)}{\left({a}–{b}\right)}$$
P(x) $$\displaystyle={\left({x}^{{2}}–{4}{x}\right)}{\left({x}^{{2}}–{4}\right)}$$
P(x) $$\displaystyle={x}^{{4}}–{4}{x}^{{3}}–{4}{x}^{{2}}+{16}{x}$$