Factoring
Def: To FACTOR a number is to express it as the product of two numbers.
Def: This product is the FACTORED FORM of the number.
Ex: 12 = 6 * 2
Def: The GREATEST COMMON FACTOR of a list of integers is the larger common
factor of those integers.
Find the GCF:
1) 30, 45
2) 72, 120, 432
3) 10, 11, 14
4) x 4 , x 5 , x 6 , x 7
Note: The exponent on a variable in the GCF is the least
exponent that appears on that variable in all the terms.
Ex:
1) 21m 7 ,18m 6 ,45m8
2) x 4 y 2 , x 7 y 5 , x 3 y 7 , y15
3)  a 2 b,ab 2
4)  12 p 5 ,18q 4
5) 12 p11 ,17q 5
Note: Negative GCFs are OKAY!
Factoring out the GCF:
In the binomial 3m  12 , there are two terms: 3m and 12. Their GCF is 3.
We then write the expression so each term is a product with 3 as one factor.
3m + 12 = 3(m) + 3(4) = 3(m + 4)
1) 5 y 2  10 y
2) 20m 3  10m 4  15m 3
3) x 5  x 3
4) 20m 7 p 2  36m 3 p 4
1
5
5) n 2  n
6
6
6) a(a  3)  4(a  3)
7) x 2 ( x  1)  5( x  1)
8) r (t  4)  5(t  4)
9) y 2 ( y  2)  3( y  2)
10) x( x  1)  5( x  1)
DISTRIBUTIVE!!!
Factoring by Grouping:
2x  6  ax  3a
Group the first two terms together and group the second two terms together. Notice the
first two terms have a GCF of 2, and the second two terms have a GCF of a.
(2 x  6)  (ax  3a )
2( x  3)  a ( x  3)
(2  a )( x  3)
Notice our factored form was the product of 2 binomials.
How can we check if our product is correct? FOIL!
Ex:
1) 6ax  24x  a  4
2) 2 x 2  10 x  3xy  15 y
3) t 3  2t 2  3t  6
4) pq  5q  2 p  10
5) 2 xy  3 y  2 x  3
6) 2a 2  4a  3ab  6b
7) x 3  3x 2  5 x  15
Rearranging Terms Before Factoring by Grouping:
10 x 2  12 y  15 x  8 xy
2(5 x 2  6 y)  x(15  8 y)
By grouping as before, we do not obtain a factored form.
Try rearranging the terms so as to get a better factored form:
10 x 2  8 xy  12 y  15 x
2 x(5 x  4 y )  3(4 y  5 x)
2 x(5 x  4 y )  3(5 x  4 y )
(2 x  3)(5 x  4 y )
Ex:
1) 2 xy  12  3 y  8 x
2) 6 y 2  20w  15 y  8 yw
3) 9mn  4  12m  3n