Factoring Trinomials with "Who Let the Dogs Out?"

                             3x² + 4x + 1 

This equation might look scary.  You're asked to factor it and don't even know where to start.  We're going to refer to this problem as "who let the dogs out?" for our visual learners.  They need to focus on the number/digit in front of the x<sup>2  

Teachers in school normally refer to this method as slide and divide, but we are going to start looking specifically at the number in front as the number of dogs you let out of the house.</sup>

^{Kids need to review lots of equations like this one and simply tell you how many dogs they are letting out.}

7x² + 32x - 15 would involve  7 dogs getting let out.

2x² – 5x +3 would involve 2 dogs getting let out.

(Give them LOTS of examples to just LOOK at and tell you the number of dogs they let out. What we are REALLY doing, is removing that number temporarily from the equation.  We will put it back after we factor.

 3x² + 4x + 1 ,

 when you let out the three dogs, the resulting simplification looks like the one here:  

 X² + 4x + 3

  When you take the number away from the X²  , the number on the end (1) must be multiplied by the number of dogs (3). You don't need to do anything yet to the middle variable. This is what teachers refer to as sliding.  You are sliding the number to the end of the equation and multiplying it to the last number in the equation.

(Do LOTS of these).

In the equation:    7x² + 32x - 15 

  the result is :        X² +32x -105

because we multiplied -15 x 7 = -105 (remember the number with the sword (negative sign) wins. See Dark Vadar blog.

In the equation:    2x² – 5x +3

the resulting simplification is:  X² -5x + 6

because we multiplied 2 x 3 to get +6 when we let the dogs out.

I am using happy sans font to make this seem more fun.

The next steps are called " The 1, 2, 3 of Factoring."

Under the simplified equations, have the kids automatically put the two factoring signs on their papers.  We'll call them the DOG BOWLS.  Tell the kids that the x represents the number of dog bones.  Have them put a bone in each bowl.           

 

                    (x      )  ( x     )

Because we've let the dogs out, each simplified equation begins with X^{2   
      so you can just put an x in each bowl.}

^{Isn't this getting easier?}

^{Now...the 1,2,3 of Factoring.  The number that needs to be factored is the last one in the equation.} 

In this last equation:  X²-5x + 6   you start factoring 6 by making a 1,2,3 chart like this:

                                                                 

  These are the first three multiples of 6.  One of these combinations most likely will work in the factoring because it is a small number. Larger numbers will require you to go higher than the 1,2,3, but this is a good first step to remember.

 NOW...ta da... look at the middle variable.

It is -5x.  Of the three pairs, when the numbers are added or subtracted, the result must be -5.  Because it is a -5, you would choose the 1,6 combination--making a -6 and 1 as the factors. It is safe to always give the larger factor the negative sign.  When you combine those two numbers, the result is -5. 
                                (x -6) (x +1)    

In this equation: X² + 4x + 3   you would be factoring 3. 

There is only one possibility  1,3.  No need to worry about negative numbers because the middle variable is positive.
 
                         (x+3) (x+1 )               

 In the equation:  X² +32x -105, the factoring chart looks like this.  Note that 105 is not evenly divisible by 2, so it is not a factor.

                              

   Because I am looking for a combination that makes +32, I choose the third pair on the list.  35 -3 = 32.
     ( x+35)(x-3) 

Practice lots of these before letting the dogs back into the house.

You know those dogs HAVE to come back in the house.  So you need to bring the number back into the equation by dividing.  You need to divide the number in each "dog dish" or factor by the number you took out.

  In the equation  3x² + 4x + 1  the factors were (x+3) (x+1 ).  
  So we have to divide 3/3 and 1/3.  Since 3/3=1 the first bowl  and the second bowl is 1/3 (x+1) (x+1/3 ).  You can NOT have a FRACTION as a factor.  When the number is not divisible evenly, you simply put the number you divided by in front of that x in the "bowl" or factor.  So the final factors would be (x+3) (3x+1 ).  

In the equation 7x² + 32x - 15  the factors were ( x+35)(x-3).
So we have to divide the 35/7 and the -3/7 to get  ( x+35)(x-3).. You can NOT have a FRACTION as a factor, so the result is( x+5)(7x-3).

In the  equation 2x² – 5x +3 the factors were (x+3) (x+1 ).  
So we have divide the 3/2 and the 1/2 since we let two dogs out.  Since both would result in fractions the final factors would be (2x+3) (2x+1 ).

                       Who Let the Dogs OUt?  Why you did.