The following strategy for factoring trinomials is often called 'factoring by grouping' or 'factoring by decomposition'. Notice that this strategy can also be used for trinomials of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi»c«/mi»«/math». Think about why this is true.

Key Lesson Marker


Factoring «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»b«/mi»«mi»x«/mi»«mo»+«/mo»«mi»c«/mi»«/math»by Decomposition (or Factoring by Grouping)

  • Determine two integers that multiply to give ac and add to give b.

  • Split the x-term using the same two integers. This is where the term 'decomposition' comes from.

  • Separately remove the GCF from the first two terms and the last two terms. This is where the term 'grouping' comes from.

  • Remove the GCF from the resulting expression.

 

Example 1

Factor «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»11«/mn»«mi»x«/mi»«mo»+«/mo»«mn»12«/mn»«/math».

The b-value is 11 and the product ac is 24. Use inspection or a table to determine two integers with a sum of 11 and a product of 24.

The integers 3 and 8 have a product of 24 and a sum of 11, so they can be used to factor.

Recall that the sum of these values represent b, so split the b-term into 8 x and 3 x.

Next, separately remove the GCF from the first two terms and from the last two terms.

Notice that the two expressions in brackets are the same, and can therefore be factored out as a GCF.

So, . This can be verified by expanding and then simplifying.



The expression «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»11«/mn»«mi»x«/mi»«mo»+«/mo»«mn»12«/mn»«/math» can be expressed in the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»m«/mi»«mi»n«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mo»(«/mo»«mi»m«/mi»«mi»q«/mi»«mo»+«/mo»«mi»n«/mi»«mi»p«/mi»«mo»)«/mo»«mi»x«/mi»«mo»+«/mo»«mi»p«/mi»«mi»q«/mi»«/math», and then factored as shown in the previous example. The table below shows a colour-coded step-by-step procedure relating the two expressions.