Precalculus (6th Edition) Blitzer

The required ${{n}^{th}}$ degree polynomial function is ${{x}^{3}}-3{{x}^{2}}-15x+125$
The general linear equation is $f\left( x \right)={{a}_{n}}\left( x-{{c}_{1}} \right)\left( x-{{c}_{2}} \right)\left( x-{{c}_{3}} \right)$ , the roots are ${{c}_{1}}=4+3i$ , ${{c}_{2}}=4-3i$ and ${{c}_{3}}=-5$. \begin{align} & f\left( x \right)={{a}_{n}}\left( x-{{c}_{1}} \right)\left( x-{{c}_{2}} \right)\left( x-{{c}_{3}} \right) \\ & ={{a}_{n}}\left( x-4-3i \right)\left( x-4+3i \right)\left( x+5 \right) \\ & ={{a}_{n}}\left( {{x}^{3}}-3{{x}^{2}}-15x+125 \right) \end{align} Calculate the value of ${{a}_{n}}$. Use $f\left( 2 \right)=91$ \begin{align} & f\left( x \right)={{a}_{n}}\left( {{x}^{3}}-3{{x}^{2}}-15x+125 \right) \\ & 91={{a}_{n}}\left( {{\left( 2 \right)}^{3}}-3{{\left( 2 \right)}^{2}}-15\left( 2 \right)+125 \right) \\ & ={{a}_{n}}\left( 8-12-30+125 \right) \\ & {{a}_{n}}=\frac{91}{91} \\ & =1 \end{align} Substitute the value of ${{a}_{n}}$ in the linear equation. \begin{align} & f\left( x \right)={{a}_{n}}\left( {{x}^{3}}-3{{x}^{2}}-15x+125 \right) \\ & =1\left( {{x}^{3}}-3{{x}^{2}}-15x+125 \right) \\ & ={{x}^{3}}-3{{x}^{2}}-15x+125 \end{align} Therefore, the polynomial equation is ${{x}^{3}}-3{{x}^{2}}-15x+125$.