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CHAPTER 8
Section 1
Monomials and Factoring
Objectives
Example 1 Classify Numbers as Prime or Composite
Example 2 Prime Factorization of a Positive Integer
Example 3 Prime Factorization of a Negative Integer
Example 4 Prime Factorization of a Monomial
Example 5 GCF of a Set of Monomials
Example 6 Use Factors
Factor 22. Then classify it as prime or composite.
To find the factors of 22, list all pairs of whole numbers
whose product is 22.
1 * 22
Factor 31. Then classify it as prime or composite.
The only whole numbers that can be multiplied together
to get 31 are 1 and 31.
2 * 11
Answer: Since 22 has more than two factors, it is
a composite number. The factors of 22, in
increasing order, are 1, 2, 11, and 22.
Factor each number. Then classify it as prime
or composite.
a. 17
Answer: 1, 17; prime
Answer: The factors of 31 are 1 and 31. Since the
only factors of 31 are 1 and itself, 31 is a
prime number.
Find the prime factorization of 84.
Method 1
The least prime factor of 84 is 2.
84 = 2 *42
The least prime factor of 42 is 2.
84 = 2 *2* 21
84 = 2 * 2 * 3 * 7 The least prime factor of 21 is 3.
All of the factors in the last row are prime.
b. 25
Answer: Thus, the prime factorization of 84 is 1*2*2*3*7
Answer: 1, 5, 25; composite
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Find the prime factorization of 60.
Method 2 Use a factor tree.
84
21
3
84 = 21 * 4
4
7
2
2
21 = 3 * 7 and 4 = 2 * 2
All of the factors in the last branch of the factor tree
are prime.
Answer: 1*2*2*3*5 = 1*22*3*5
Answer: Thus, the prime factorization of 84 is 1*2*2*3*7
Or
1*22 * 3 * 7
Find the prime factorization of –132.
Express –132 as –1 times 132.
Find the prime factorization of –154.
Answer:
Answer: The prime factorization of –132 is
or
Factor
completely.
Factor
completely.
Express –26 as –1 times 26.
=1•2•3•3•x•x•x•y•y•y
Answer:
9 = 3• 3
in factored form is 1•2•3•3•x•x•x•y•y•y
Answer:
in factored form is
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Find the GCF of 12 and 18.
Factor each monomial completely.
Factor each number.
a.
Circle the common prime factors.
Answer:
1•3•5•a•a•a•b•b
The integers 12 and 18 have one 2 and one 3 as
common prime factors. The product of these common
prime factors,
or 6, is the GCF.
b.
Answer: The GCF of 12 and 18 is 6.
Answer: -1•3•3•5•x•y•y
.
Find the GCF of
Answer: The GCF of
Find the GCF of each set of monomials.
Factor each number.
a. 15 and 35
Circle the common prime factors.
Answer: 5
and
is
.
b.
and
Answer:
Rene has crocheted 32 squares for an afghan. Each
square is 1 foot square. She is not sure how she will
arrange the squares but does know it will be rectangular
and have a ribbon trim. What is the maximum amount of
ribbon she might need to finish
an afghan?
Find the factors of 32 and draw rectangles with each
length and width. Then find each perimeter.
The greatest perimeter is 66 feet. The afghan with this
perimeter has a length of 32 feet and a width of 1 foot.
The factors of 32 are 1, 2, 4, 8, 16, 32.
Answer: The maximum amount of ribbon Rene will need
is 66 feet.
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Mary wants to plant a rectangular
flower bed in her front yard with a
stone border. The area of the flower
bed will be 45 square feet and the
stones are one foot square each. What
is the maximum number of stones that
Mary will need to go around all four
sides of the flower bed?
Answer: 92 feet
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