Department of Mathematics

Polynomials and Factoring

Worked Examples

Some Special Formulas

The formulas below are useful when factoring or expanding polynomial expressions.
Difference of squares: $X^2-Y^2=(X+Y)(X-Y)$
Perfect square (sum): $(X+Y)^2=X^2+2XY+Y^2$
Perfect square (difference): $(X-Y)^2=X^2-2XY+Y^2$s
Cube of a sum: $(X+Y)^3=X^3+3X^2Y+3XY^2+Y^3$
Cube of a difference: $(X-Y)^3=X^3+3X^2Y-3XY^2-Y^3$
Sum of cubes: $X^3+Y^3=(X+Y)(X^2-XY+Y^2)$
Difference of cubes: $X^3-Y^3=(X-Y)(X^2+XY+Y^2)$

Example. Factor $9y^2-12y+4$.

Rewriting the expression, $$9y^2-12y+4=(3y)^2-2(2)(3y)+(2)^2$$ We see that it fits the perfect square formula using $3y$ and $2$, so we get: $$9y^2-12y+4=(3y-2)^2$$.