# 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 $x=1+2\sqrt{3}$.

• Why did you ask the question in the first place when you already knew the answer? – Roby5 May 15 '16 at 18:36
• @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. – John Wayland Bales May 15 '16 at 21:10

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