Find all the factors of 6x
3
Solution:
The factors of 6x
3
are 1,x,x
2
, x
3
and 2,2x,2x
2
,2x
3
and 3,3x,3x
2
,3x
3
and 6,6x,6x
2
,6x
3
Example 8:
Find all the factors of 4y
2
Solution:
The factors of 4y
2
are 1,y,y
2
and 2,2y,2y
2
and 4,4y,4y
2
The greatest common factor (GCF) of two expressions is the largest factor they both share.
There are two methods to finding the GCF of two expressions the first is by writing out all the
factors of each expression and then finding the largest result that they both share. This method is
very time consuming as the following worked example shows.
Example 9:
Find the GCF of 12x
3
and 16x
2
Solution:
The factors of 12x
3
are
1
,
x
,
x
2
, x
3
2,2x,2x
2
,2x
3
3,3x,3x
2
,3x
3
4,4x,4x
2
,4x
3
6,6x,6x
2
,6x
3
12,12x,12x
2
,12x
3
The factors of 16x
2
are
1
,
x
,
x
2
2,2x,2x
2
4,4x,4x
2
8,8x,8x
2
16,16x,16x
2
The common factors are
1,x,x
2
2,2x,2x
2
4,4x,
4x
2
The GCF of 12x
3
and 16x
2
is 4x
2
The second method for finding the GCF of two expressions is to find the GCF of the constants
and the exponentials separately and then combine the result.
Example 10:
Find the GCF of 12x
3
and 16x
2
Solution:
The GCF of 12 and 16 is 4
The GCF of x
2
and x
3
is x
2
(The smaller of the two powers)
The GCF of 12x
2
and 16x
3
is 4x
2
Example 11:
Find the GCF of 10y
4
x
2
and 25y
3
x
3
Solution:
The GCF of 10 and 25 is 5
The GCF of x
2
and x
3
is x
2
(The smaller of the two powers)
The GCF of y
4
and y
3
is y
3
(The smaller of the two powers)
The GCF of 10y
4
x
2
and 25y
3
x
3
is 5x
2
y
3