Find the complex zeros of the following polynomial function. Write f in factored form. f(x)=x^3-12x^2+49x-58

Emeli Hagan 2021-09-16 Answered
Find the complex zeros of the following polynomial function. Write f in factored form.
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}-{12}{x}^{{2}}+{49}{x}-{58}\)
The complex zeros of f are =?

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Expert Answer

Szeteib
Answered 2021-09-17 Author has 5271 answers
Step 1
Given: \(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}-{12}{x}^{{2}}+{49}{x}-{58}\)
It can be written as,
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{3}}-{2}{x}^{{2}}-{10}{x}^{{2}}+{20}{x}+{29}{x}-{58}\)
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}{\left({x}-{2}\right)}-{10}{x}{\left({x}-{2}\right)}+{29}{\left({x}-{2}\right)}\)
\(\displaystyle{f{{\left({x}\right)}}}={\left({x}-{2}\right)}{\left({x}^{{2}}-{10}{x}+{29}\right)}\)
Step 2
Roots of f(x) or zeroes of f(x) are given by \(\displaystyle{\left({x}-{2}\right)}{\left({x}^{{2}}-{10}{x}+{29}\right)}\)
\(\displaystyle\Rightarrow{x}={2}\ \text{ of }\ {x}^{{2}}-{10}{x}+{29}={0}\)
\(\displaystyle{x}={2}\ \text{ of }\ {x}={\frac{{{10}\pm\sqrt{{{100}-{4}{\left({29}\right)}}}}}{{{2}}}}={5}\pm{2}{i}\) (complex zeroes)
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