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Unit 9: Factoring
Lesson 1: Introduction to Factoring
Factors and Multiples
6 ● 5 = 30
6 ● 8 = 48
_________
_______
___ and ____ are _________ of _____.
___ and ____ are _________ of _____.
_________ is a multiple of ________.
_________ is a multiple of ________.
_________ is a multiple of ________.
_________ is a multiple of ________.
Common Factors
If two numbers are divisible by the same number, then they are called common factors
___ is a common factor of ______ and ________.
Try These on Your Own
1. List the factors of 72.
2. 36 is divisible by _____.
3. _____ is a multiple of 12.
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
4. ______ is a common factor of 32 & 40.
Unit 9: Factoring
Prime and Composite Numbers
Prime Numbers
Composite Numbers
Greatest Common Factor
The Greatest Common Factor is the greatest integer that is a common factor
of two or more numbers.
Example 1
Example 2
What is the greatest common factor for 18
and 27?
What is the greatest common factor for
18
27
3x and 9x2
9x2
3x
Example 3
Example 4
What is the greatest common factor of
What is the greatest common factor of
6x2 y and 9x3 y2 ?
2x and 3y?
6x2 y
9x3 y2
2x
3y
If: 2x + 3y
This polynomial is ____________ because
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
Lesson 1: Introduction to Factoring Practice
Part 1: Read each question and answer completely.
1. 35 is divisible by _______ (give 2 solutions)
2. Name 6 factors of 12.
3. ________ is a multiple of 13. (Give 3 solutions)
4. _______ is a common factor of 36 and 48.
5. ______ is a common factor of 144 and 120.
6. ______ is the greatest common factor of 48 and 64.
7. ______ is the greatest common factor of 48 and 60.
8. Is 9 a prime number? Explain why or why not.
9. 15 is a composite number. Explain why.
10. Do you think that the number 1 is prime, composite, or neither? Explain.
Part 2: Find the greatest common factor (GCF) for each set of numbers.
1. 12a2 and 14 a
7. 3xy,
2x2y,
2. x2 y3 and 2xy2
8. 3a2b, 27ab2, and 9ab
3. 3ab5 and 4a3b3
9. 2x2y3z,
4. 9rs and 12r2s
10. 15rs3t, 7r2st, and 5r2s2t
5. 4a3b2 and 2a3
11. 18xz, 9x2y4z2, and 3xyz
6. 16x4yz2 and 24x3z
12. 7x2y5, 5x5y4, and 8x3y5
and
4x3y
6xy2z2, and 18xz3
Find the greatest common factor (GCF) for each set of numbers.
(2 points each)
1. 13x3yz and 26xy2z2
2. 18r5s8t4, 81r6s5t6, and 27r9s7t5
3. 22abc and 11a2b
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
Lesson 1: Introduction to Factoring Practice – Answer Key
Part 1: Read each question and answer completely.
1. 35 is divisible by 7 and 5 (give 2 solutions)
2. Name 6 factors of 12.
(7 & 5 are factors of 35)
(1,2,3,4,6,12)
(1 x12) (2 x6) (3x4)
3. 26, 39, 52 is a multiple of 13. (Give 3 solutions) (13 x2 = 26) (13x3 = 39) (13 x4 = 52) (You may
also have had: 65, 78, 91, 104, 117, 130 …)
4. 6 is a common factor of 36 and 48. (6 x6 = 36) (6 x 8 = 48) (You may also have had 4 because
(4 x9 = 36) (4 x 12 = 48). Other answers include: 2 , 3, and 12
5. 12 is a common factor of 144 and 120. (12 x12 = 144) (12 x 10 = 120) (Other answers include:
2, 3, 4, 6, 8, 24)
6. 16 is the greatest common factor of 48 and 64.
Factors of 48:
Factors of 64:
1, 2, 3, 4,6, 8, 12, 16, 24 ,48
1, 2, 4, 8, 16, 32, 64
The common factors are: 1,2,4, 8, 16
The greatest common factor would be 16.
7. 12 is the greatest common factor of 48 and 60.
Factors of 48:
Factors of 60:
1, 2, 3, 4,6, 8, 12, 16, 24 ,48
1,2,3,4,5,6,10,12,15,20,30,60
The common factors are: 1,2,3,4,6, 12
The greatest common factor would be 12.
8. Is 9 a prime number? Explain why or why not.
9 is not a prime number. A prime number only has factors of 1 and itself. 9 has the following factors:
1,3,9. Since 3x3=9, 9 is not a prime number.
9. 15 is a composite number. Explain why.
15 is a composite number. A composite number has one or more factors other than itself and 1. (It’s the
opposite of a prime number.)
The factors of 15 are: 1,3,5,15. Since 3x5=15, we can classify 15 as a composite number.
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
10. Do you think that the number 1 is prime, composite, or neither? Explain.
Since the only factor of 1 is 1, it is not considered to be prime or composite.
The prime numbers less than 50 are:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47.
Part 2: Find the greatest common factor (GCF) for each set of numbers.
1. 12a2 and 14 a
GCF: 2a
The GCF for 12 and 14 is 2.
The GCF for a2 and a is a
2. x2 y3 and 2xy2
7. 3xy,
2x2y,
and
4x3y
GCF: xy
The GCF for 3,2, and 4 is 1. (We don’t
need to write)
The GCF for x, x2 and x3 is x.
The GCF for y, y, and y is y
8. 3a2b, 27ab2, and 9ab
GCF: xy2
GCF: 3ab
The GCF for 1 and 2 is 1. (We don’t
need to write.)
The GCF for x2 and x is x.
The GCF for y3 and y2 is y2
The GCF for 3, 27, and 9 is 3.
The GCF for a2 and a is a.
The GCF for b, and b2 is b.
3. 3ab5 and 4a3b3
9. 2x2y3z,
6xy2z2, and 18xz3
GCF: ab3
GCF: 2xz
The GCF for 3 and 4 is 1. (We don’t
need to write.)
The GCF for a3 and a is a.
The GCF for b5 and b3 is b3.
The GCF for 2, 6 and 18 is 2.
The GCF for x2 and x is x.
Since 18xz3 does not have a y term, we
cannot include a y in our GCF.
The GCF for z3, z2 and z is z.
4. 9rs and 12r2s
10. 15rs3t, 7r2st, and 5r2s2t
GCF: 3rs
GCF: rst
The GCF for 9 and 12 is 3.
The GCF for r2 and r is r.
The GCF for s is s.
The GCF for 15, 7 and 5 is 1. (We don’t
need to write.)
The GCF for r2 and r is r.
The GCF for s3,s2, and s is s.
The GCF for t is t.
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com
Unit 9: Factoring
5. 4a3b2 and 2a3
11. 18xz, 9x2y4z2, and 3xyz
GCF: 2a3
GCF: 3xz
The GCF for 4 and 2 is 2.
The GCF for a3 is a3.
Since 2a3 does not have a b term, we
cannot include a b term in the GCF.
The GCF for 18, 9 and 3 is 3.
The GCF for x2 and x is x.
Since 18xz does not have a y term, we
cannot include a y term in the GCF.
The GCF for z2 and z is z.
6. 16x4yz2 and 24x3z
12. 7x2y5, 5x5y4, and 8x3y5
GCF: 8x3z
GCF: x2y4
The GCF for 16 and 24 is 8.
The GCF for x4 and x3 is x3.
Since 24x3z does not have a y term, we
cannot include a y term in our GCF.
The GCF for z2 and z is z.
The GCF for 7 and 5, and 8 is 1. (We
don’t need to write.)
The GCF for x2, x3, and x5 is x2.
The GCF for y4 and y5 is y4.
1. 13x3yz and 26xy2z2
2. 18r5s8t4, 81r6s5t6, and 27r9s7t5
GCF: 13xyz
GCF: 9r5s5t4
The GCF for 13 and 26 is 13.
The GCF for x3 and x is x.
The GCF for y2 and y is y.
The GCF for z and z2 is z.
The GCF for 18, 81 and 27 is 9.
The GCF for r5, r6 and r9 is r5.
The GCF for s8, s5 and s7 is s5.
The GCF for t4, t6, and t5 is t4.
3. 22abc and 11a2b
GCF: 11ab
The GCF for 22 and 11 is 11.
The GCF for a2 and a is a.
The GCF for b is b.
Since 11a2b does not have a c term, we
cannot include a c term in our GCF.
Copyright © 2009-2010 Karin Hutchinson – Algebra-class.com