Exercise: Evaluate a polynomial function such as $P(x)=2x^3-3x^2+7x-2$ at a surd such as $x=1+2\sqrt{3}$. Exercise: Given polynomial function $P(x)=2x^3-3x^2+7x-2$ evaluate $P(x)$ at the surd $x=1+2\sqrt{3}$.

Solution: Divide $P(x)$ by the divisor
\begin{equation}
D(x)=[x-(1+2\sqrt{3})]\cdot[x-(1-2\sqrt{3})]=x^2-2x-11
\end{equation}
Then 
\begin{equation}
P(x)=(2x+1)(x^2-2x-11)+31x+9
\end{equation}
Therefore
\begin{equation}
P(1+2\sqrt{3})=0+31(1+2\sqrt{3})+9=40+62\sqrt{3}
\end{equation}
