Question

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

Polynomials
ANSWERED
asked 2021-02-25
Find a polynomial of the specified degree that has the given zeros. Degree 4, zeros -2, 0, 2, 4

Answers (1)

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) \(\displaystyle={x}^{{4}}–{4}{x}^{{3}}–{4}{x}^{{2}}+{16}{x}\)
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