Given: n=3. 4 and 2i are zeros. f(-1)=75 . Find an nth-degree polynomial function with real coefficients

Tabansi 2021-09-09 Answered

Given:
\(n=3\).
4 and 2i are zeros.
\(f(-1)=75\)
Find an nth-degree polynomial function with real coefficients

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Expert Answer

2abehn
Answered 2021-09-10 Author has 13313 answers

\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}-{4}\right)}{\left({x}-{2}{i}\right)}{\left({x}+{2}{i}\right)}\)
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}-{4}\right)}{\left({x}^{{2}}-{2}{i}{x}+{2}{i}{x}+{4}\right)}\)
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}-{4}\right)}{\left({x}^{{2}}+{4}\right)}\)
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left[{x}^{{2}}{\left({x}-{4}\right)}+{4}{\left({x}-{4}\right)}\right]}\)
\(\displaystyle{f{{\left({x}\right)}}}={a}{\left({x}^{{3}}-{4}{x}^{{2}}+{4}{x}-{16}\right)}\)
Use \(\displaystyle{f{{\left(-{1}\right)}}}={75}\) to solve for a:
\(\displaystyle-{75}={a}{\left[{\left(-{1}\right)}^{{3}}-{4}{\left(-{1}\right)}^{{2}}+{4}{\left(-{1}\right)}-{16}\right]}\)
\(-75=-25a\)
\(a=3\)
\(\displaystyle{f{{\left({x}\right)}}}={3}{x}^{{3}}-{12}{x}^{{2}}+{12}{x}-{48}\)

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