Factoring Trinomials: Factor Grid Method

Let's talk about factoring trinomials. This method is exhaustive.

Procedure

Ax²+Bx+C

Step 1: List out factors of A and C. Factors of A form the rows and factors of C form the columns.

Step 2:  Setup the grid. Multiply the column by the row and put result in the corresponding square. It forms major squares and minor squares.

Step 3:  Look at the diagonals of each major square.

•  If C is positive, then find the sum resulting in B. (Outer + Inner)
•   If C is negative, then find the difference resulting in B. (Outer-inner or inner-outer)

Step 4: Setup the factors by using the grid.

• First, label the diagonal of the major square with inner and outer.
•  Second, label the row term (FI – first inner) and column term (LI- last inner) that line up with the “inner” term of the diagonal.
• Third, label the row term (FI – first outer) and column term (LI- last outer) that line up with the “outer” term of the diagonal.
• Fourth, using (FO_LI)(FI_LO) setup the factors.

Step 5: Choose the signs.

Mechanics

Binomial times a binomial. It involves a method called FOILing.

(A+B)(C+D)

AC+AD+BC+DB

First+Outer+Inner+Last

On the grid:

First/Last	B	D
A		AD=Outer
C	BC=Inner

Therefore, one can set up their factors this way.

First/Last	B - LI	D - LO
A –FO		AD=Outer
C – FI	BC=Inner

(By simply seeing how they align. C is part of both the first term and inner term, A is part of both the outer term and the first term, etc.)

(FO+LI)(FI+LO)

The other diagonal produces another possible:

(C+B)(A+D)

AC+CD+BA+DB

First+Outer+Inner+Last

On the grid

First/Last	B	D
A	BA=Inner
C		CD=Outer

First/Last	B - LI	D - LO
A –FI	BA=Inner
C –FO		CD=Outer

(By simply seeing how they align. A is part of both the first term and inner term, C is part of both the outer term and the first term, etc.)

(FO+LI)(FI+LO)

Notice on the grids. One grid can contain two possible combinations of the outer and inner while the first and the last remain the same.

Examples

1. x²+5x+6

Step 1: List out factors of A and C

A=1

1x*1x

C=6

3*2, 6*1

Step 2: Setup the grid

First/Last	2	3	6	1
1x	2x	3x	6x	1x
1x	2x	3x	6x	1x

Step 3: Look at the diagonals of each major square.

First/Last	2	3	6	1
1x	2x	3x	6x	1x
1x	2x	3x	6x	1x

Step 4:

First/Last	2 -LO	3 - LI	6	1
1x - FO	2x - outer	3x	6x	1x
1x - FI	2x	3x -inner	6x	1x

Using (FO+LI)(FI+LO), you get (x_3)(x_2)

Step 5:

(x+3)(x+2)

2. 3x²-2x-16

Step 1:

A=3

3x*1x

C=16

1*16, 2*8, 4*4

Step 2: Setup the grid

First/Last	1	16	2	8	4	4
3x	3x	48x	6x	24x	12x	12x
1x	1x	16x	2x	8x	4x	4x

Step 3: Look at the diagonals of each major square.

First/Last	1	16	2	8	4	4
3x	3x	48x	6x	24x	12x	12x
1x	1x	16x	2x	8x	4x	4x

Step 4:

First/Last	1	16	2 - LI	8 - LO	4	4
3x – FI	3x	48x	6x – inner	24x	12x	12x
1x - FO	1x	16x	2x	8x – outer	4x	4x

Using (FO+LI)(FI+LO), you get (x_2)(3x_8)

Step 5:

(x+2)(3x-8)

3. 8x²+114x+81

Step 1:

A=8

8x*1x, 4x*2x

C=81

1*81, 3*27, 9*9

Step 2: Setup the grid

First/last	1	81	3	27	9	9
8x	8x	648x	24x	216x	72x	72x
1x	1x	81x	3x	27x	9x	9x
4x	4x	324x	12x	108x	36x	36x
2x	2x	162x	6x	54x	18x	18x

Step 3: Look at the diagonals of each major square.

First/last	1	81	3	27	9	9
8x	8x	648x	24x	216x	72x	72x
1x	1x	81x	3x	27x	9x	9x
4x	4x	324x	12x	108x	36x	36x
2x	2x	162x	6x	54x	18x	18x

Step 4:

First/last	1	81	3 LI	27 LO	9	9
8x	8x	648x	24x	216x	72x	72x
1x	1x	81x	3x	27x	9x	9x
4x FO	4x	324x	12x	108x- outer	36x	36x
2x FI	2x	162x	6x - inner	54x	18x	18x

(FO_LI)(FI_LO)=(4x_3)(2x_27)

Step 5:

(4x+3)(2x+27)

5. 4x²-81 (special type – Difference of Squares)

Step 1:

A=4

4x*1x, 2x*2x

C=81

1*81, 3*27, 9*9

Step 2: Setup the grid

First/last	1	81	3	27	9	9
4x	4x	324x	12x	108x	36x	36x
1x	1x	81x	3x	27x	9x	9x
2x	2x	324x	6x	54x	18x	18x
2x	2x	162x	6x	54x	18x	18x

Step 3: Look at the diagonals of each major square.

First/last	1	81	3	27	9	9
4x	4x	324x	12x	108x	36x	36x
1x	1x	81x	3x	27x	9x	9x
2x	2x	324x	6x	54x	18x	18x
2x	2x	162x	6x	54x	18x	18x

Notice: With a difference of squares, it is the major square that has the same term in each box. This happens because the middle term is 0x.

Step 4:

First/last	1	81	3	27	9 - LI	9 -LO
4x	4x	324x	12x	108x	36x	36x
1x	1x	81x	3x	27x	9x	9x
2x - FI	2x	324x	6x	54x	18x –inner	18x
2x - FO	2x	162x	6x	54x	18x	18x -outer

(FO_LI)(FI_LO)=(2x_9)(2x_9)

Step 5:

(2x-9)(2x+9)