Overview

Polynomials appear often at AIME level. Learn how to relate roots and coefficients and how to build polynomials with given roots.

Key Ideas

• If roots are $r_1, r_2$, then $x^2-(r_1+r_2)x+r_1r_2=0$.
• If $p(a)=0$, then $(x-a)$ is a factor of $p(x)$.
• Symmetric expressions often reduce via Vieta.

Worked Example

If $r$ and $s$ are roots of $x^2-5x+6=0$, compute $r^2+s^2$.

We have $r+s=5$ and $rs=6$. Then $r^2+s^2=(r+s)^2-2rs=25-12=13$.

Practice Problems