Overview

Advanced algebra focuses on structure: factor clever expressions, transform systems, and use symmetry.

Key Ideas

Core Skills

Use Symmetric Substitution

If an expression is symmetric in $x$ and $y$, set $s=x+y$ and $p=xy$ to reduce degree.

Factor Strategically

Check for standard identities (difference of squares, sum/difference of cubes) before expanding.

Complete the Square

Rewriting quadratics as $(x-h)^2+k$ helps with bounds and root structure.

Worked Example

Solve $x+y=5$ and $x^2+y^2=13$.

Compute $2xy=(x+y)^2-(x^2+y^2)=25-13=12$, so $xy=6$. Then $t^2-5t+6=0$, giving $t=2$ or $3$. So $(x,y)$ is $(2,3)$ or $(3,2)$.

More Examples

Example 1: Vieta and Symmetry

If $x+y=4$ and $xy=1$, find $x^3+y^3$.

$x^3+y^3=(x+y)^3-3xy(x+y)=64-12=52$.

Example 2: Completing the Square

Find the minimum of $x^2-6x+13$.

$x^2-6x+13=(x-3)^2+4$, so the minimum is $4$.

Example 3: System Reduction

Solve $x+y=6$ and $xy=5$.

$t^2-6t+5=0$ gives $t=1$ or $5$, so $(x,y)=(1,5)$ or $(5,1)$.

Strategy Checklist

• Look for symmetry and use $s=x+y$, $p=xy$.
• Factor before expanding.
• Complete the square for extrema or root structure.

Practice Problems

• Expanding expressions that factor cleanly.
• Dropping solutions when converting to $s$ and $p$.
• Forgetting to check back in the original system.