Transcript Monomials and Factoring

Monomials and Factoring
Section 8 – 1
Main Ideas
 Find prime factorizations of integers
and monomials.
 Find the greatest common factors of
integers and monomials.
Prime Factorization :
 When two or more numbers are
multiplied, each number is called a
factor of the product.
Vocabulary
Definition
A prime number is a whole
Prime
number, greater than 1, whose
Number
Ex.
5
only factors are 1 and itself.
Composite A composite number is a whole
number, greater than 1, that has
Number
10
more than two factors.
Prime
Factorization
Prime factorization occurs when a 45 =
2(5)
3
whole number is expressed as a
product of factors that are all
prime numbers.
Example 1: Factor each number. Then
classify each number as prime or composite.
a.) 28
List all the pairs of whole numbers whose product is 28.
1(28) = 28
2(14) = 28
4(7) = 28
Therefore, the factors of 28 are {1, 2, 4, 7, 14, 28}.
Since 28 has more than 2 factors it is a composite
number.
b.) 31
 List all the pairs of whole numbers whose
product is 31.
 1(31) = 31
 Therefore, the factors of 31 are {1, 31}.
 Since the only factors of 31 are itself and
1, it is a prime number.
You can also do a factor tree
200
= 2∙100
= 2 ∙10 ∙10
= 2 ∙ 2∙ 5 ∙2 ∙ 5
All of these numbers are prime so the
prime factorization of 200 is
2 5
3
2
Try:
Find the factors of each number. Then
classify the number as prime or composite.
1.) 41
2.) 121
3.) 90
Find the prime factorization of each integer
1.) 600
2.) 175
3.) -150
Find the factors of each number. Then
classify the number as prime or
composite.
1.) 41
1(41) = 41
Therefore the factors of 41 are {1, 41}.
It is a prime number.
Find the factors of each number. Then
classify the number as prime or
composite.
2.) 121
1(121) = 121
11(11) = 121
The factors of 121 are {1, 11, 121}.
Therefore it is a composite number.
Find the factors of each number. Then
classify the number as prime or
composite.
3.) 90
Therefore the factors of 90
1(90) = 90
are
2(45) = 90
{1, 2, 3, 5, 6, 9, 10, 15, 18,
3(30) = 90
30, 45, 90}
5(18) = 90
It is a composite number.
6(15) = 90
9(10) = 90
Find the prime factorization of each
integer
1.) 600
2(300) = 600
2(2)(150) = 600
2(2)(2)(75) = 600
2(2)(2)(3)(25) = 600
2(2)(2)(3)(5)(5) = 600
2  3 5
3
2
Find the prime factorization of each
integer
2.) 175
5(35) = 175
5(5)(7) = 175
5 7
2
Find the prime factorization of each
integer
3.) -150
-1(150) = -150
-1(2)(75) = -150
-1(2)(3)(25) = -150
-1(2)(3)(5)(5) = -150
 1 2  3  5
2
Example: Find the GCF of each set of
monomials.
2(2)(3) = 12
12 and 18
2(3)(3) = 18
Factor each number
Circle the common
2(6) = 12
prime factors
2(2)(3) = 12
2(9) = 18
2(3)(3) = 18
The GCF of
12 and 18 are
2(3) = 6
1.) Find the GCF of 12 and 48
2(6) = 12
2(2)(3) = 12
2(24) = 48
2(2)(12) = 48
2(2)(2)(6) = 48
2(2)(2)(2)(3) = 48
2(2)(3) = 12
2(2)(2)(2)(3) = 48
The GCF of
12 and 48 is
2(2)(3) = 12
2.) Find the GCF of 44 and 100
2(22) = 44
2(2)(11) = 44
2(50) = 100
2(2)(25) = 100
2(2)(5)(5) = 100
2(2)(11) = 44
2(2)(5)(5) = 100
The GCF of 44 and 100
is (2)(2) = 4
3. Find the GCF of 10 and 21
Example: Factor the monomial completely
32x2
2(16)(x)(x) = 32x2
2
2(2)(8)(x)(x) = 32x
2
2(2)(2)(4)(x)(x) = 32x
2
2(2)(2)(2)(2)(x)(x) = 32x
Example
16xy2z2 and 72xyz2
2(8)(x)(y)(y)(z)(z) = 16xy2z2
2(2)(4)(x)(y)(y)(z)(z) = 16xy2z2
2(2)(2)(2)(x)(y)(y)(z)(z) = 16xy2z2
2(36)(x)(y)(z)(z) = 72xyz2
2(2)(18)(x)(y)(z)(z) = 72xyz2
2(2)(2)(9)(x)(y)(z)(z) = 72xyz2
2(2)(2)(3)(3)(x)(y)(z)(z) = 72xyz2
Example
2(2)(2)(2)(x)(y)(y)(z)(z) = 16xy2z2
2(2)(2)(3)(3)(x)(y)(z)(z) = 72xyz2
The GCF of 16xy2z2 and 72xyz2 is
2(2)(2)(x)(y)(z)(z) = 8xyz2
Try: Find the GCF of each set of
monomials
1.) 49x and 343x2
2.) 18a3b2 and 36a3b2
1.) 49x and 343x2
(7)(7)(x) = 49x
7(49)(x)(x) = 343x2
7(7)(7)(x)(x) = 343x2
(7)(7)(x) = 49x
7(7)(7)(x)(x) = 343x2
The GCF is
(7)(7)(x) = 49x
2.) 18a3b2 and 36a3b2
2(9)(a)(a)(a)(b)(b) = 18a3b2
2(3)(3)(a)(a)(a)(b)(b) = 18a3b2
2(18)(a)(a)(a)(b)(b) = 36a3b2
2(2)(9)(a)(a)(a)(b)(b) = 36a3b2
2(2)(3)(3)(a)(a)(a)(b)(b) = 36a3b2
2(3)(3)(a)(a)(a)(b)(b) = 18a3b2
2(2)(3)(3)(a)(a)(a)(b)(b) = 36a3b2
The GCF is 2(3)(3)(a)(a)(a)(b)(b) = 18a3b2
(Do you understand each direction
and the difference between them?)
Find the GCF of each set of
monomials
Factor each monomial
completely
Find the GCF of each set of
monomials.
Assignment
page 422 (1-16, 19-27, 37-42)