The GCF for a polynomial expression will not always be a monomial. In the next example, the GCF of the expression is a binomial.

 

Example 3

Factor «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«mo»)«/mo»«mo»(«/mo»«mn»4«/mn»«mo»)«/mo»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#8201;«/mo»«mo»(«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«mo»)«/mo»«mo»(«/mo»«mi»x«/mi»«mo»)«/mo»«/math».

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»2«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/math» is a common factor to both «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»2«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/math» and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»2«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«/math» . There are no factors common to 4 and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»x«/mi»«/math», other than 1, so «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»2«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/math» is the GCF.

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn mathcolor=¨#B94A48¨»2«/mn»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/math» can be moved outside a set of brackets using the distributive property in reverse.

«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#1C82BA¨»2«/mn»«mi mathcolor=¨#1C82BA¨»x«/mi»«mo mathcolor=¨#1C82BA¨»+«/mo»«mn mathcolor=¨#1C82BA¨»1«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»+«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#1C82BA¨»2«/mn»«mi mathcolor=¨#1C82BA¨»x«/mi»«mo mathcolor=¨#1C82BA¨»+«/mo»«mn mathcolor=¨#1C82BA¨»1«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»§#160;«/mo»«mo mathcolor=¨#B94A48¨»=«/mo»«mo mathcolor=¨#B94A48¨»§#160;«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#1C82BA¨»2«/mn»«mi mathcolor=¨#1C82BA¨»x«/mi»«mo mathcolor=¨#1C82BA¨»+«/mo»«mn mathcolor=¨#1C82BA¨»1«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mo mathcolor=¨#B94A48¨»+«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»)«/mo»«/math»