Factoring: Which
method?
The following is a checklist based on
the number of terms.
Factoring a binomial (two terms)
Remove any common factor.
Check for the difference of squares:
x2 – 9
( x + 3 )( x – 3 )
Example
32 a4 -2
Remove the common factor of 2
2 ( 16 a4 - 1 )
Difference of squares
2 ( 4 a2 + 1 ) ( 4 a2 -1 )
Difference of squares again
2 ( 4a2 + 1 )( 2a + 1 )( 2a- 1)
A sum or difference of cubes is also a possibility:
x3 + y3 = ( x + y )( x2- xy + y2 )
x3 - y3 = ( x - y )( x2 + xy + y2 )
Factoring a trinomial (three terms) when
coefficient of x2 is 1
the
Remove any common factor.
Check for a factorable trinomial. Common patterns are:
x2 + x + #
x4 + x2 + #
x2 + xy + y2
x4 + x2y2+y4
etc.
To factor a trinomial, find factors of the product of the coefficients of the first and
last terms which give a sum matching the coefficient of the middle term.
Example:
x2 + 4x - 21
Factors of -2l whose sum is 4 are 7 and -3:
( x + 7 )( x - 3 )
Expand ( x + 7 )( x - 3 ) to make sure it verifies.
Factoring four terms
Remove any common factor.
Try grouping two pairs of terms. (Sometimes terms need to be rearranged.)
Remove common factor.
3ax - 5x2 + 3ay - 5xy
Example:
No common factor to be removed.
Let's group the first two and last two terms:
x ( 3a - 5x ) + y ( 3a - 5x )
Remove common factor of ( 3a- 5x ):
( 3a - 5x ) ( x + y )
More on factoring a trinomial when
the
coefficient of x2 is not 1
Remove any common factor.
Check for a factorable trinomial. Common patterns are:
x2 + x + #
x4 + x2 + #
x2 + xy + y2
x4 + x2y2 + y4 etc.
To factor a trinomial, find factors of the product of the coefficients of the first and
last terms that give a sum matching the coefficient of the middle term.
Example
-12 x2 - 46x - 40
Remove common factors:
-2( 6x2 + 23x + 20 )
This trinomial has a factorable pattern.
Find factors of 6 * 20 = 120,
whose sum is 23
1 and 120
121
2 and 60
62
3 and 40
43
4 and 30
34
5 and 24
29
6 and 20
26
8 and 15
23
Replace the middle term with 8x and 15 x
-2( 6x2 + 15x + 8x + 20)
Group into pairs and remove common factors:
-2 ( 3x (2x + 5) + 4 (2x + 5))
Remove the common factor, ( 2x + 5 ):
-2 ( 2x +5 ) ( 3x + 4 )
Example:
-8a2 - 2ab + 3b2
Remove common factors:
- ( 8a2 + 2ab - 3b2 )
This trinomial has a factorable pattern.
Find factors of 8 * - 3 = -24, whose sum is 2
-1 and 24
23
-2 and 12
10
-3 and 8
5
-4 and 6
2
Replace the middle term with -4ab and 6ab:
Group into pairs and remove common factors:
Remove the common factor, (2a – b)
Keep an eye out for Perfect Squares:
- ( 8a2 - 4ab + 6ab – 3b2 )
- ( 4a(2a - b) + 3b (2a - b))
-( 2a – b )( 4a + 3b )