C. Factoring Polynomials using the Greatest Common Factor

A polynomial can be factored using the GCF of its terms. Because the GCF is a factor of all of the terms, the distributive property can be used in reverse to show the GCF multiplied by a simpler polynomial factor.

For example, «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mn mathcolor=¨#1C82BA¨»5«/mn»«msup mathcolor=¨#1C82BA¨»«mi mathcolor=¨#1C82BA¨»x«/mi»«mn»2«/mn»«/msup»«mi mathcolor=¨#1C82BA¨»y«/mi»«mo»)«/mo»«mo»(«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«mo»)«/mo»«mo»+«/mo»«mo»(«/mo»«mn mathcolor=¨#1C82BA¨»5«/mn»«msup mathcolor=¨#1C82BA¨»«mi mathcolor=¨#1C82BA¨»x«/mi»«mn»2«/mn»«/msup»«mi mathcolor=¨#1C82BA¨»y«/mi»«mo»)«/mo»«mo»(«/mo»«mn»2«/mn»«mi»y«/mi»«mo»)«/mo»«mo»-«/mo»«mo»(«/mo»«mn mathcolor=¨#1C82BA¨»5«/mn»«msup mathcolor=¨#1C82BA¨»«mi mathcolor=¨#1C82BA¨»x«/mi»«mn»2«/mn»«/msup»«mi mathcolor=¨#1C82BA¨»y«/mi»«mo»)«/mo»«mo»(«/mo»«mn»4«/mn»«mo»)«/mo»«/math» has a GCF of «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mi»y«/mi»«/math» and can be rewritten as «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#1C82BA¨»5«/mn»«msup mathcolor=¨#1C82BA¨»«mi mathcolor=¨#1C82BA¨»x«/mi»«mn»2«/mn»«/msup»«mi mathcolor=¨#1C82BA¨»y«/mi»«mo»(«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»y«/mi»«mo»-«/mo»«mn»4«/mn»«mo»)«/mo»«/math», using the distributive property in reverse.

Example 1

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»16«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»20«/mn»«mi»x«/mi»«/math» using the GCF of its terms.

Determine the GCF.

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mn mathcolor=¨#B94A48¨»4«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»3«/mn»«/msup»«/mstyle»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»=«/mo»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mn»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mi»x«/mi»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«/mstyle»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»16«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«/mstyle»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»=«/mo»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn mathcolor=¨#B94A48¨»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn mathcolor=¨#B94A48¨»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mi»x«/mi»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«/mstyle»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»20«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«/mstyle»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»=«/mo»«/math»
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn»2«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«mo mathcolor=¨#B94A48¨»§#183;«/mo»«mi»x«/mi»«/mstyle»«/math»

The GCF is «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«/math».

Write each term as the product of the GCF and another monomial factor.

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mn mathcolor=¨#B94A48¨»4«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»3«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»16«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»20«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mstyle»«/math»

Use the distributive property in reverse.

 «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mn mathcolor=¨#B94A48¨»4«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»3«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»16«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»20«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«/mstyle»«/math»  «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mrow»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mrow»«/mstyle»«/math»
   «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»(«/mo»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mstyle»«/math»
 


Factorization can be checked by multiplying the factors. If a polynomial was factored correctly, the product of its factors should equal the original polynomial expression.

Example 2

Verify that «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mi»x«/mi»«mo»(«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«mi»x«/mi»«mo»-«/mo»«mn»5«/mn»«mo»)«/mo»«/math» is equivalent to «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»-«/mo»«mn»16«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»20«/mn»«mi»x«/mi»«/math».

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»(«/mo»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mstyle»«/math»   «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mrow»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»5«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mrow»«/mstyle»«/math»
   «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mo mathcolor=¨#B94A48¨»=«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«msup mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»3«/mn»«/msup»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»16«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»20«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«/mstyle»«/math»