# Find the complex zeros of the following polynomial function. Write f in factored form.f(x)=2x^4-5x^3-20x^2+115x-52. The complex zeros of f are ?

Find the complex zeros of the following polynomial function. Write f in factored form.
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{4}}-{5}{x}^{{3}}-{20}{x}^{{2}}+{115}{x}-{52}$$
The complex zeros of f are ?

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Fatema Sutton

Step 1
Factoring polynomials
Step 2
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{4}}-{5}{x}^{{3}}-{20}{x}^{{2}}+{115}{x}-{52}$$
$$\displaystyle={2}{x}^{{4}}+{8}{x}^{{3}}-{13}{x}^{{2}}-{52}{x}^{{2}}+{32}{x}^{{2}}+{128}{x}-{13}{x}-{52}$$
$$\displaystyle={2}{x}^{{3}}{\left({x}+{4}\right)}-{13}{x}^{{2}}{\left({x}+{4}\right)}+{32}{x}{\left({x}+{4}\right)}-{13}{\left({x}+{4}\right)}$$
$$\displaystyle={\left({x}+{4}\right)}{\left({2}{x}^{{3}}-{13}{x}^{{2}}+{32}{x}-{13}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({2}{x}^{{3}}-{x}^{{2}}-{12}{x}^{{2}}+{6}{x}+{26}{x}-{13}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({x}^{{2}}{\left({2}{x}-{1}\right)}-{6}{x}{\left({2}{x}-{1}\right)}+{13}{\left({2}{x}-{1}\right)}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({2}{x}-{1}\right)}{\left({x}^{{2}}-{6}{x}+{13}\right)}$$
Real zeros at $$\displaystyle{x}=-{4},{\frac{{{1}}}{{{2}}}}$$
Comple zeros are root of $$\displaystyle{x}^{{2}}-{6}{x}+{13}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({2}{x}-{1}\right)}{\left({x}^{{2}}-{6}{x}+{9}+{4}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({2}{x}-{1}\right)}{\left({\left({x}+{3}\right)}^{{2}}+{4}\right)}$$
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}+{4}\right)}{\left({2}{x}-{1}\right)}{\left({x}+{3}+{2}{i}\right)}{\left({x}+{3}-{2}{i}\right)}$$
Now complex zeros are $$\displaystyle{x}=-{3}+{2}{i}{\quad\text{and}\quad}-{3}-{2}{i}$$