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Name_____________________________________________Date_________Period __________
Greatest Common Factor (GCF)
Definition: In a multiplication expression, the quantities being multiplied are called the FACTORS.
Example: 42  6  7
Example: 3x  3  x
therefore 6 and 7 are FACTORS of 42.
therefore 3 and x are the FACTORS of 3x .
A number or expression can be factored into lowest terms. This means taking the non-prime factors
to the lowest number possible.
Example 1: 42  6  7  3  2  7
Therefore
3  2  7 is considered lowest terms because 3, 2, and 7 are all
prime numbers and cannot be simplified any lower.
the factored form of 42 is 3  2  7 .
Example 2: Factor 56 into lowest terms. (Sometimes a factor tree helps!!!) 56
56  7  8
7  8
56  7  2  4
56  7  2  2  2
Final Answer
2  4
2  2
Example 3: Factor 2x y into lowest terms.
3
2x 3 y  2  x 3  y
2x3y = 2  x  x  x  y
Final Answer
If the number does not have any factors other than itself and 1, then the number is said
to be PRIME.
Example: Factor 13 into lowest terms.
13  13  1
Therefore 13 is prime.
Factor the following expressions to lowest terms.
1. 33
___________________
2. 40 _______________________
3. 100
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4
4. a b _______________________
2 2
5. 50n m
___________________
Find the missing factor.
6.
y 5  ?y 3 
2
8. 7x  ? 7x 
 
_________
7
3
7. c  ? c
_________
9.

 6p 5  ? 6p 4
_________

_________
The GREATEST COMMON FACTOR (GCF) is the greatest number that is a factor of all the numbers.
Example 4: Find the GCF of 21 and 15.
21  3  7
15  3  5
3 is the biggest number that they have in COMMON therefore the GCF is 3.
Example 5: Find the GCF of w2z4 and u2w4.
w 2z 4  w  w  z  z  z  z
2
2
The common terms are w  w  w , therefore w is the GCF.
u 2w 4  u  u  w  w  w  w
Example 6: Find the GCF of 18 and 24.
18  2  3  3
24  2  2  2  3
Each term has a 2 & a 3 in common therefore the GCF is 2  3  6
Find the greatest common factor.
10. 28, 42
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12. 75a3, 125a2
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11. 55uv2, 65u2w2
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13. 120a3b2, 90ab3
__________
When factoring polynomials, the first step is to find the GCF between the 2 terms.
Example 7: 5a + 35
GCF of 5a and 35 is
5
Once you have the GCF, the GCF gets “taken out” of the expression and the factors that are “left”
stay in parentheses as an expression.
GCF
5•a
5•7
5 (a+7)
“left over”
from 1st term
therefore 5(a + 7) is the solution
“left over”
from 2nd term
A sure way of knowing if your answer is correct is to use the distributive property.
5(a + 7) = 5a + 35 
**Factoring polynomials is just like doing the distributive property backwards!!
Example 8: 4x + 6y
GCF =
Check: ___________________
Example 9: -42gh - 30h2
GFC =
Check: __________________
3 4 5
3 4
2 2 2
Example 10:  42a b x  56a x  168a b x
GCF =
Factor.
14. -6c2 - 3
15. 4k3r2q - k2rq2 - 10kq
16. 108c2n2z5 - 35c4nz4
17. 3k – 12k2g + 30k5
18. Barbara has a rectangular lettuce garden. The area of the garden is (25x2 + 15x – 5x3).
What is the length if the width is (5x + 3 – x2)?
19. Jessica was paid $(-6x2p2 + 30xp5 – 15x5p3) for working a week at Kroger. If she was paid an
hourly rate of $(-3xp2) per hour, how many hours did she work?