# Factoring

Definition: An expression is said to be factored if it is written as a product of 2 or more factors. So factoring is the opposite of applying the distributive law.

Samples of factoring

4th Grade and up: Taking out a common factor

 Expression Factored 3x+9 3(x+3) -2y+46xy -2y(1-23x) or   2y(23x-1) 5x-25y+10 5(x-5y+2) 5x-27y cannot be factored (a-7a) a(1-7)=a(-6)=-6a

7th Grade & up: Reducing expressions by factoring

 Expression Factored Reduced \frac{3x+9}{3} \frac{3(x+3)}{3} x+3 \frac{-2y+46xy}{y^2} \frac{-2y(1-23x)}{y^2} \frac{-2(1-23x)}{y} \frac{5x-25y+10}{x-5y} \frac{5(x-5y+2)}{x-5y} cannot be reduced \frac{5x-27y}{54y-10x} \frac{5x-27y}{2(27y-5x)} = \frac{5x-27y}{-2(5x-27y)} \frac{1}{-2}= -\,\frac{1}{2}

 Expression with a>0 and b>0 Factored InterActivity/Mathcast a^2-b^2 (a-b)(a+b) a^2+b^2 cannot be factored a^2+2ab+b^2 (a+b)(a+b)=(a+b)^2 Interactivity a^2-2ab+b^2 (a-b)(a-b)=(a-b)^2
 Rule: A quadratic binomial or trinomial in x can be factored only if the graph (the parabola) crosses the x-axis (has real roots).