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}

8th Grade & up: Factoring special quadratic binomials and trinomials

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).

8th Grade & up: Factoring quadratic trinomials

Expression	Factored	InterActivity/Mathcast
x^2 +7x+12 c	(x+3)(x+4)	Mathcast demo (first)
x^2 -x+15 c	Has no whole number factorization	Mathcast demo (last)

Standards for 4th-6th grade: CA 5.AF.1.3, CA 6.AF.1.3, ACT Number Sense 20-23

Standards for 7th grade & up: CA 7.AF.1.3, ACT Number Sense 24-27

Standards for 8th grade & up: ACT Expressions, Equations & Inequalities 24-28, Algebra 1: 5-1, Algebra 1: 5-6, ACT Expressions, Equations & Inequalities 28-32