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Quadratic Factoring

To factor a quadratic expression written in the general form of ax² + bx + c,  means to factor a trinomial into two binomials . The binomials are expressions that when multiplied using the distributive property will result in the trinomial. A trinomial has three terms, an x² term, an x term, and a constant. After factoring, the expression will show the multiplication of two binomials, each with two terms that are added or subtracted.

To break down a trinomial the FOIL method is used in reverse. The steps in the FOIL method are:

First – Multiply the first term in each set of parentheses

Outer – Multiply the outer term in each set of parentheses

Inner – Multiply the inner term in each set of parentheses

Last  – Multiply the last term in each set of parentheses

 Example 1: x² + 5x + 6

 Set up parentheses to show the 2 binomials like this:

 

( __ + ___ )( ___ + ____)

 

Start by listing the factors of the first and last term. 5x will be the trickiest one to find so it will be last.

 

Factors of x² : x

Factors of 6: -6,-3,-2,-1,1,2,3,6

The only way to get x² is to multiply x by itself.

 

( x__ + ___ )( __x_ + ____)

The next step is to find two factors of 6 that add up to 5 for the term 5x. If the terms in the binomials were negative the middle term would also be negative so we need a positive pair of factors that add up to 5. The only possible factors that fit are 2 and 3.

 

( x__ + __2_ )( __x_ + _3___)

Now check using the FOIL method.

 

( x__ + __2_ )( __x_ + _3___)

 

First: (x)(x)= x²

Outside: (x)(3)=3x

Inside: (2)(x)=2x

Last: (2)(3)=6

x² +3x + 2x + 6= x² +5x+6 This is the original trinomial. All the terms are positive in this example but this is not always the case.

 

Example 2:

x² -6x +8

Factors of x²: x

Factors of 8: -8,-4,-2,-1,1,2,4,8

The second term is negative so the 2 factors must be negative. Two negative integers added together will equal a negative and multiplied will equal a positive.

( x__ -__2_ )( __x_ -_4___)

Use FOIL to check:

First: (x)(x)= x²

Outside: (x)(-4)=-4x

Inside: (-2)(x)=-2x

Last: (-2)(-4)=8

 

x² -4x – 2x + 8= x² -6x+8

 

Example 3:

x² -x -12

Factors of x²: x

Factors of 12: -12,-6,-4,-3,-2,-1,1,2,3,4,6,12 The constant in this example is negative so it must be the product of a negative integers and a positive integer. The 2 factors must add up to -1, the b value.  -4 and 3 are the only factors that add up to -1 while also resulting in -12 when multiplied.

( x__ -__3_ )( __x_ +_4___) 

 

 

 

 

Did paraphrasing the words help you internalize the concepts more?

Paraphrasing was very helpful because I needed to clearly picture the steps before I could translate them into language my students would understand. I also had to create my own examples and then apply the steps of both factoring and then checking using FOIL to make sure they were accurate examples. That let me think through the process so that I was better prepared to teach this as a lesson.

How can you apply this type of exercise in a lesson for your own students?

Students could create a demonstration for teaching factoring to students in another class using technology such as PowerPoint or iMovie where they can visually show the steps. They could then describe the development process as well as the presentation in their interactive math notebooks or in a journal.

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