Polynomials and Factoring

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: $$9y^2-12y+4=(3y-2)^2$$.