Factoring by Grouping
with a Grid Box
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for this lesson a will NOT be 1.

Let's look at factoring 10x2 - 11x - 6.
This time, let's use our grouping process written in a grid box.


When "factoring by grouping", dealing with the signs may be problematic.  To avoid this confusion, many students find the creation of a "grid box" to be helpful.

1. We start, again, by multiply the leading coefficient, a, and the constant term, c.
         
10 (-6) = -60
 
2.
We still consider all of the possible factors of this new product and we find the one pair that adds to the middle term's coefficient, b.  For this example, we need a sum of -11.

                         4 + -15 = -11

Factors of -60:
(1) (-60)
(2) (-30)
(3) (-20)
(4) (-15)
(5) (-12)
(6) (-10)
(10) (-6)
(12) (-5)
(15) (-4)
(20) (-3)
(30) (-2)
(60) (-1)

3.
Now, we prepare a "gird box" that will hold the terms.
10x2  
  -6

Start by placing the leading term in the upper left hand corner and the constant term in the lower right hand corner of the grid.

10x2 -15x
4x -6

As we did in grouping, split the middle term into two new terms using your found factors.  Order is not important for the split terms, as either arrangement works.

 

4.
 
     5x
10x2 -15x
4x -6
     +2

Now, find a common factor in each
 ROW.

 

 

        2x          -3
         5x
10x2 -15x
4x -6
        +2

Find the common factor in each
COLUMN.

5.

 

Notice that the answer is now found on the edges of the grid box.
                                       (5x + 2)(2x -3 ANSWER:

Warning!  This grid box approach ONLY works if you have factored out any common factors BEFORE beginning the grid.
See example below.

Notice what happens in this example which contains a common factor of 2,
that has not been factored out before starting the process.

Factor by grouping:

4x - 2x - 6
4 -6 = -24
factors -6 and +4 add to middle term -2
4x - 6x + 4x - 6
(4x - 6x) + (4x - 6)
2x(2x - 3) + 2(2x - 3)
(2x + 2)(2x - 3)
Now, factor further.
2(x + 1)(2x - 3) ANSWER
 

Even though we did not FACTOR out the common factor before starting the standard "factoring by grouping" method, we were still able to arrive at the correct answer (by factoring the final answer).

Factor by grouping w/grid box:

4x - 2x - 6
4 -6 = -24
factors -6 and +4 add to middle term -2

        4x          -6
          2x
4x2 -6x
4x -6
         +2

(2x + 2)(4x - 6)  Factor further.
2(x + 1)2(2x - 3) 
4(x + 1)(2x - 3) OOPS!  Wrong!

If we use the "grid method" when we have not factored the common factor BEFORE starting the process, we get the WRONG answer.

Be very careful to factor out common factors FIRST, if you use this box method!