Find a degree 3 polynomial having zeros -6, 6 and 0 and leading coefficient equal to 1

Find a degree 3 polynomial having zeros -6, 6 and 0 and leading coefficient equal to 1

Question
Functions
asked 2021-02-12
Find a degree 3 polynomial having zeros -6, 6 and 0 and leading coefficient equal to 1

Answers (1)

2021-02-13
If c is a zero of a polynomial, then x−c is a linear factor of this polynomial.
Given the zeros −6, 6, and 0, we can write:
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}+{6}\right)}{\left({x}−{6}\right)}{\left({x}\right)}\)
Expand:
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}^{{{2}}}−{36}\right)}{\left({x}\right)}\)
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}^{{{3}}}−{36}{x}\right)}\) Since the leading coefficient is equal to 1, we simply let a=1:
\(\displaystyle{f{{\left({x}\right)}}}={1}{\left({x}^{{{3}}}−{36}{x}\right)}\)
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{3}}}−{36}{x}\)
0

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