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The Greatest common factor (GCF) of a polynomial is the product of
- The largest number that divides evenly into the coefficients
- The smallest exponent of any variable common to all terms.
One method to find the Greatest Common Factor (GCF)
between two integers is to list all the factors of the
integer and choose the largest factor they have in
common. (There are other methods that can be used
to find a GCF.)
The GCF of 24 and 18 equals 6
The GCF of 9 and 20 equals 1
Find the greatest common
factor
4
2
2
2
3
4
5
2
6
3
4
2
Solution
The product of the smallest exponent of any variable common to all
terms
4
2
GCF 𝑥
2
2
2
3
GCF 𝑥𝑦
2
4
5
2
6
4
(both terms do not have a z, there is no z in the GCF
3
4
3
2
𝑇ℎ𝑒 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 𝑖𝑠 1 , 𝑠𝑜 𝑡ℎ𝑒 𝑝𝑎𝑟𝑒𝑛𝑡ℎ𝑒𝑠𝑖𝑠 𝑖𝑠𝑛
′
𝑡 𝑛𝑒𝑒𝑑𝑒𝑑.
Find the Greatest common factor
2
, 12 𝑥
Solution:
2
, 12 𝑥
2 is the largest number that divides evenly into
10 and 12.
The coefficient of the GCF is 2
x is the smallest exponent of the common
variable x.
The variable part of the GCF is x
Answer: 𝐺𝐶𝐹 = 2 𝑥
Find the GCF
3
2
𝑦, 6 𝑥𝑦
Solution
3
2
𝑦, 6 𝑥𝑦
2 is the largest number that divides evenly into
14 and 8 and 6.
The coefficient of the GCF is 2
x is the smallest exponent of the common
variable x.
y has the smallest exponent of the common
variable y
The variable part of the GCF is 𝑥𝑦
Answer: 𝐺𝐶𝐹 = 2 𝑥𝑦
Rewrite by factoring out the GCF
2
Step 1: Find the GCF
Step 2: Write the GCF in front of a
parenthesis and place the original
problem in the parenthesis with
each term divided by the GCF
Step 3 : Simplify (This is the
answer)
Step 4 : Check
Solution
2
Step 1: Find the GCF
3 is the largest number that divides evenly into both 15
and 24.
The number part of the GCF is 3.
The x terms are 𝑥 𝑎𝑛𝑑 𝑥
2
the 𝑥
1
term has the lowest
exponent.
The variable part of the GCF is x.
Step 2: Write the GCF in front of a parenthesis and place
the original problem in the parenthesis with each term
divided by the GCF
15 𝑥
2
3 𝑥
24 𝑥
3 𝑥
Step 3: Simplify (This is the answer)
Step 4: Check
2
Rewrite by factoring out the GCF
3
4
2
3
2
Step 1: Find the GCF
Step 2: Write the GCF in front of a
parenthesis and place the original
problem in the parenthesis with each
term divided by the GCF
Step 3: Simplify (This is the answer)
Step 4: Check
Solution
3
4
2
3
2
Step 1: Find the GCF
9 is the largest number that divides evenly into both
27, 18 and 36.
The number part of the GCF is 9.
Each term has a x, and 𝑥
1
term has the lowest
exponent.
Each term has a y, and 𝑦
2
term has the lowest
exponent.
The variable part of the GCF is 𝑥𝑦
2
GCF= 9 𝑥𝑦
2
Step 2: Write the GCF in front of a parenthesis and
place the original problem in the parenthesis with
each term divided by the GCF
2
27 𝑥
3
𝑦
4
9 𝑥𝑦
2
18 𝑥
2
𝑦
3
9 𝑥𝑦
2
36 𝑥𝑦
2
9 𝑥𝑦
2
Step 3: Simplify (This is the answer)
2
2
2
Step 4: Check
2
2
2
2
2
2
2
2
3
4
2
3
2
Rewrite by factoring out the GCF
Step 1: Find the GCF
Step 2: Write the GCF in front of a
parenthesis and place the original
problem in the parenthesis with each
term divided by the GCF
Step 3: Simplify (This is the answer)
Step 4: Check
Step 1: 1 is the largest number that divides evenly into 1
and 12 and 7.
The coefficient of the GCF is 1
Only the first term has a single x. There is no 𝑥
𝑝𝑜𝑤𝑒𝑟
term
in the GCF.
Both terms have an (𝑥 − 5 ) each has an exponent of 1.
The variable part of the GCF is (𝑥 − 5 ) Keep this
parenthesis, it will be helpful.
GCF = (𝑥 − 5 )
Step 2: (𝑥 − 5 ) (
12 (𝑥− 5 )
(𝑥− 5 )
7 (𝑥− 5 )
(𝑥− 5 )
Step 3: Cancel the (𝑥 − 5 )
Answer:
( 𝑥 − 5
)( 12 𝑥 + 7
)
Step 4: To check I need to rewrite the original problem and
rewrite my answer to show they equal each other.
Original problem rewrite:
2
2
Answer Rewrite: (𝑥 − 5 )( 12 𝑥 + 7 )
= 𝑥
( 12 𝑥
)
( 7
)
( − 5
)( 12 𝑥
)
( − 5
)( 7
)
2
2
The original problem and my answer equal when
simplified, so my answer is correct. ✓
#1- 16 : Factor out the GCF.
3
2
3
2
5
4
3
3
2
4
3
3
2
2
3
3
3
4
4
2
4
2
3
2
6
7
3
2
3
3
2
3
2
4
3
3
2
4
3
2
2
#17- 26 : Factor out a (−) from each polynomial.
#27- 38 : Factor each polynomial by factoring out the opposite of the GCF.
3
2
2
3
2
5
3
4
3
3
2
4
2
2
2
2
3
5
2
3
2
2
3
3
4
3
3
2
2
5
2
3
2
