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

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    $\begingroup$ Why did you ask the question in the first place when you already knew the answer? $\endgroup$ – Roby5 May 15 '16 at 18:36
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    $\begingroup$ @Roby5 I asked and answered my own question because I could not find this method explained on Stack Exchange and I hoped it would be instructive to others searching Stack Exchange. From my experience teaching pre calculus for 45 years I have many times found students trying to evaluate polynomial functions at surds by using synthetic division and have made it a practice of showing them an easier way. I believe this is a permitted practice. $\endgroup$ – John Wayland Bales May 15 '16 at 21:10
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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}

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