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

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Page last modified on January 06, 2009, at 01:08 PM