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```Factoring by GCF
Remember factor Trees?? Create a factor tree for the following two numbers.
60
24
6 * 10
4 * 6
2*3 * 2*5
2*2 * 2*3
Are there any numbers common to each?? Sure!
When we break the numbers down using factor
trees, we can see the common factors. In the
example to the left,
60 = 2·2·3·5
24 = 2·2·2·3
The common parts are 2, 2, and 3. If we multiply
these, we get 2·2·3 = 12. So, 12 is the Greatest
Common Factor (GCF).
Now, think through the same process using the following sets of numbers:
60
60 =2·2·3·5
75
75 = 3·5·5
45
Break each number down into factors….
45 = 3·3·5 Now, pick the factors that are in all three…
3·5 = 15
The Greatest Common Factor is 15.
We can do the same with monomials!! Find the Greatest Common Factor:
24x 3 y2
= 2·2·2·3·x·x·x·y·y
56x 2 y4
Break each into factors….
= 2·2·2·7·x·x·y·y·y·y Now pick the factors that are in both…
2·2·2·x·x·y·y
The GCF = 8x 2 y2
Summarize: What do you notice about the powers of the variables in the
GCF of the two monomials above?? The GCF variables have the lowest
power for the variable from the monomials! (What is the lowest power of
the x for the monomials? x 2 What is the lowest power of the y for the
monomials? y 2 These are the powers of the variables in the GCF)
When you pull the Greatest Common Factor out of all terms of a polynomial and place the left
over parts inside of a parenthesis, it is called Factoring by GCF:
24x 3 y2 + 56x 2 y4 = 8x 2 y2 ( 3x + 7y 2 ) This is called reverse distributive or factoring by GCF
Here is another example to factor 60 x3 y 2 z 4  24 xy3 z 3
First find the GCF for 60 x 3 y 2 z 4 and 24 xy3 z 3 ………….. 12 x1 y 2 z 3 is the GCF!!!
60 x 3 y 2 z 4  24 xy3 z 3 factors as 12 xy2 z 3 (5x 2 z  2 y)
That means:
More examples:
3x + 6xy – 9 x 2
Factors into
3x(1 + 2y - 3 x )
 m 2  5m  5
Factors into
-( m 2  5m  5 ) ……. Try to leave highest power term +
9x + 6y
Factors into
3(3x + 2y)
Problems to try: Factor by GCF.
1. 40 x 5  24 x 3 y
2.  18x 3 y 2  45x 2 y  9 xy
3. 32 x 5  48x 3 y  80 xy 2
4. 64 x 4  16 x 3
5.  x12 8x 5  12 xy
6. 36 x 5 y 3 z  18x 4 yz  42 xyz  24 xz