Finding the GCF of Algebraic Expressions

Name ———————————————————————
CHAPTER
4
Date ————————————
Finding the GCF of Algebraic
Expressions
For use after Lesson 4.2
You have learned to ﬁnd the GCF of numbers by writing the prime factorization of
each number and then multiplying the common factors. A similar procedure is used
to ﬁnd the GCF of expressions with variables.
EXAMPLE 1
Find the GCF of x 3 and x 2y.
Solution
Step 1: Expand each expression by writing as a product.
x3 x x x
x2y x x y
Step 2: Identify the common variables in both expressions.
x
x
xx xy
Step 3: Multiply the common variables to ﬁnd the GCF.
x
xx
2
The GCF of x3 and x2y is x2. 䊏
EXAMPLE 2
Find the GCF of m 5n 2, m 3n, and m 2n 2.
Step 1: Expand each expression by writing as a product.
m5n2 m m m m ⴢ m ⴢ n ⴢ n
m3n m m m n
m2n2 m m n n
Step 2: Identify the common variables in the expressions.
mmmnn
n
m
nn
m m
m m
m m
Step 3: Multiply the common variables.
m
mnmn
2
The GCF of m5n2, m3n, and m2n2 is m2n. 䊏
Guided Practice for Examples 1 and 2
Find the GCF.
1. a4b2 and ab5
2. s4tu2 and st 3u
3. x 5y, x 4y 2, and x 3y 2
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Solution
351
Name ———————————————————————
CHAPTER
4
Date ————————————
Finding the GCF of Algebraic
Expressions continued
For use after Lesson 4.2
EXAMPLE 3
Find the GCF of 12x 4yz and 18xyz.
Solution
Step 1: Expand the expressions using prime factorization and writing the variables as
products.
12x 2yz 2
18xyz 2
23xxyz
33xyz
Step 2: Identify the common factors and variables.
2
2
23 xxyz
3 3 x y z
Step 3: Multiply the common factors and variables.
2
3 x y z 6xyz
The GCF of 12x 4yz and 18xyz is 6xyz.
䊏
Guided Practice for Example 3
Find the GCF.
EXAMPLE 4
5. 10c3d and 15cd 2
Find the GCF of 7(x ⴙ 3) and 8(x ⴙ 3).
Solution
Grouped expressions can be common factors.
Step 1: Expand by writing each expression as a product.
7(x 3) 7
8(x 3) 8
(x 3)
(x 3)
Step 2: Identify the common terms in each expression.
(x 3)
8 (x 3)
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352
The GCF of 7(x 3) and 8(x 3) is (x 3). 䊏
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4. 8a3b and 12a
Name ———————————————————————
CHAPTER
4
Date ————————————
Finding the GCF of Algebraic
Expressions continued
For use after Lesson 4.2
EXAMPLE 5
Find the GCF of (x ⴙ 2)(x ⴙ 7) and (x ⴙ 2)(x ⴙ 4).
Solution
Step 1: Expand by writing each expression as a product.
(x 2)(x 7) (x 2)
(x 2)(x 4) (x 2)
(x 7)
(x 4)
Step 2: Identify the common terms in each expression.
(x 2)
(x 2)
(x 7)
(x 4)
The GCF of (x 2)(x 7) and (x 2)(x 4) is (x 2). 䊏
Guided Practice for Examples 4 and 5
Find the GCF.
6. 5(m 5) and 7(m 5)
7. ( y 3)( y 4) and ( y 8)( y 4)
Find the GCF.
1. p5 and p2
2. ab and a4
3. xy 3 and x3y
4. dfe4 and d 2e3
5. s2t 2 and rst
6. j 3k 2 and j 3k
7. 3xy and 6x2
8. gh4 and 3g 3
9. 8b3 and 10b3
10. 2j 2k 5 and 3jk 2
11. 12f 3 and 16f
12. 20s5t 8 and 30s2
13. h5, h3j 3, and j 4
14. 3xy, 6x2y, and 6x
15. 14e2, 21e3, and 14e3
16. 2( y 3) and 6( y 3)
17. 8(a 1) and 20(a 1)
18. (b 5) and (b 5)
19. (v 4)(v 6) and (v 6)(v 1)
20. ( f 2)( f 2) and ( f 2)( f 5)
21. 12(x2 3) and (x 3)(x2 3)
22. 5(x 9), 10(x 9), and (x 9)(x 4) 23. 24, 12(s 7), and 8(s 7)(s 2)
24. Multiple Choice Which pair of expressions has a GCF of 2c?
A. 2c2 and 4c3
C. 4c2 and 8c
B. 4c2 and 6c
D. 8c2 and 16c4
25. Writing How is ﬁnding the GCF of a pair of numbers similar to ﬁnding the GCF
of a pair of algebraic expressions? How is it different?
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Practice
353
Name ———————————————————————
CHAPTER
4
Date ————————————
Finding the GCF of Algebraic
Expressions continued
For use after Lesson 4.2
26. Error Analysis Describe and correct the error made in ﬁnding the GCF of
15(a 6)(a 6) and 10(a 6).
15(a 6)(a 6): 3 5 (a 6)
10(a 6): 2 5 (a 6)
The GCF is 5(a 6)(a 6).
(a 6)
✗
27. Short Answer Bailey Jo and Tyler are making greeting cards for their family
and friends. Bailey Jo has x3 happy face stickers and Tyler has x2y frog stickers
and they want to use up all their stickers.
a. Write an expression for the greatest number of greeting cards they can make if
all the cards have the same number of each type of sticker.
b. Suppose x 3 and y 2. How many stickers does each person have? What is
the greatest number of greeting cards they can make? Does your answer agree
with your answer to part (a)? Explain.
28. Michelle has 3x 2y 3 cups of peanuts, 6x 4 cups of dried fruit, and 12xy 2 cups of
chocolate pieces to make trail mix for her hiking group. If she puts the same
number of cups of each ingredient into each snack bag, how many snack bags can
she make?
29. Open-Ended Math Write three algebraic expressions that have a GCF of 3x 3.
Pre-AP Copymasters
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30. Challenge Find the GCF of 24w3x 3y 5z 2, 36wxz 5, 48w 2xz, and 54y10z 4.
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