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.

Core Skills

Use the Factor Theorem

Test small integer roots quickly and factor out $(x-a)$ when $p(a)=0$.

Build a Polynomial

If roots are given, form $\prod (x-r_i)$ and expand or leave factored as needed.

Use Vieta for Symmetric Sums

Compute $r^2+s^2$ or $1/r+1/s$ using $r+s$ and $rs$.

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 $r^2+s^2=(r+s)^2-2rs=25-12=13$.

More Examples

Example 1: Polynomial from Roots

Find the monic polynomial with roots $2$ and $-3$.

$(x-2)(x+3)=x^2+x-6$.

Example 2: Reciprocal Sum

If $r+s=4$ and $rs=3$, find $\frac{1}{r}+\frac{1}{s}$.

$\frac{r+s}{rs}=4/3$.

Example 3: Factor Theorem

If $p(x)=x^3-4x^2-x+4$, show that $x=1$ is a root and factor $p(x)$.

$p(1)=0$, so $(x-1)$ is a factor. Then $p(x)=(x-1)(x^2-3x-4)$.

Strategy Checklist

• Test small integer roots first.
• Use Vieta for symmetric expressions.
• Keep polynomials factored when possible.

Practice Problems