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

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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Szeteib
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)