Find all the real and complex zeros of the following polynomials.f(x)=x^4-4x^3+7^2-16x +12

Maiclubk 2021-09-15 Answered
Find all the real and complex zeros of the following polynomials.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{4}}-{4}{x}^{{3}}+{7}^{{2}}-{16}{x}+{12}\)

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Expert Answer

Neelam Wainwright
Answered 2021-09-16 Author has 7115 answers

It is given that, \(\displaystyle{f{{\left({x}\right)}}}={x}^{{4}}-{4}{x}^{{3}}+{7}^{{2}}-{16}{x}+{12}\)
We have to find all the real and complex zeros of the polynomial.
We have, \(\displaystyle{f{{\left({x}\right)}}}={x}^{{4}}-{4}{x}^{{3}}+{7}{x}^{{2}}-{16}{x}+{12}\)
For the zeros of polynomial,
\(\displaystyle\rightarrow{x}^{{4}}-{4}{x}^{{3}}+{7}{x}^{{2}}-{16}{x}+{12}={0}\)
\(\displaystyle\Rightarrow{\left({x}-{3}\right)}{\left({x}^{{3}}-{x}^{{2}}+{4}{x}-{4}\right)}={0}\)
\(\displaystyle\Rightarrow{\left({x}-{3}\right)}{\left({x}-{1}\right)}{\left({x}^{{2}}+{4}\right)}={0}\)
\(\displaystyle\Rightarrow{\left({x}-{3}\right)}={0}\) or \(\displaystyle{\left({x}-{1}\right)}={0}\) or \(\displaystyle{\left({x}^{{2}}+{4}\right)}={0}\)
\(\displaystyle\Rightarrow{x}={3}\) or \(\displaystyle{x}={1}\) or \(\displaystyle{x}^{{2}}=-{4}\)
\(\displaystyle\Rightarrow{x}={3}\) or \(\displaystyle{x}={1}\) or \(\displaystyle{x}=\pm{2}{i}\)
There are four zeros of the polynomial
Two real numbern \(x=1,3\)
Two complex number \(\displaystyle={2}{i},-{2}{i}\)

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