Download Lesson 38

LESSON 38
Simplifying
Expressions Using the
GCF
REMEMBER:
GCF = Greatest Common Factor
Prime: a number with only 2 factors 1 and itself
Composite: a number with more then two factors.
Prime Factorization: Writing a number as a product of
primes.
PRIME FACTORIZATION:
To find the prime factorization, divide the number up until all you have is
primes.
7560
7560 = 756 x 10
7560 = (252 x 3) x (5 x 2)
7560 = 126 x 2 x 3 x 5x2
7560 = 9 x 14 x 2x3x5x2
7560 = 3 x 3 x 2 x 7 x2x3x5x2
7560 = 2x2x2x3x3x3x5x7
7560 = 23 𝑥 33 𝑥 5 𝑥 7
GREATEST COMMON FACTOR OF NUMBERS:
What is the Greatest Common Factor of 36 and 96?
Prime Factorization of 36 = 3x3x2x2
Prime Factorization of 96 = 3x2x2x2x2x2
Greatest Common Factor is what is in both: 3x2x2 = 12
PRIME FACTORIZATION OF AN ALGEBRAIC
EXPRESSION:
Find the Prime Factorization of 24𝑥 3 𝑦 2
24𝑥 3 𝑦 2 = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦
GCF OF AN ALGEBRAIC EXPRESSION
4 3
3
What is the GCF of 12𝑥 𝑦 𝑧 and 216𝑥 𝑦 ?
12𝑥 4 𝑦 3 𝑧 = 2 ∙ 2 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑧
216𝑥 3 𝑦 = 2 ∙ 3 ∙ 7 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦
3
In both we have 2 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 = 6𝑥 𝑦
FACTORING USING GCF
Factor the given polynomial: 9𝑥 3 + 6𝑥 2 + 3𝑥
Find the GCF of each term:
9𝑥 3 = 3 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥
6𝑥 2 = 2 ∙ 3 ∙ 𝑥 ∙ 𝑥
3𝑥 = 3 ∙ 𝑥
GCF of the terms is 3x
3𝑥 3𝑥 2 + 2𝑥 + 1
EXAMPLE:
What is the GCF of 24𝑚3 𝑛4 + 32𝑚𝑛5 𝑝?
24𝑚3 𝑛4 = 2 ∗ 2 ∗ 2 ∗ 3 ∗ 𝑚 ∗ 𝑚 ∗ 𝑚 ∗ 𝑛
32𝑚𝑛5 𝑝 = 25 ∗ 𝑚 ∗ 𝑛 ∗ 𝑛 ∗ 𝑛 ∗ 𝑛 ∗ 𝑛 ∗ 𝑝
In both we have: 2 ∗ 2 ∗ 2 ∗ 𝑚 ∗ 𝑛 = 8𝑚𝑛
EXAMPLE:
Factor:
𝟏𝟖𝒙+𝟒𝟓𝒙𝟑
𝟗𝒙
𝟏𝟖𝒙 = 𝟐 ∗ 𝟑 ∗ 𝟑 ∗ 𝒙
𝟒𝟓𝒙𝟑 = 𝟓 ∗ 𝟑 ∗ 𝟑 ∗ 𝒙 ∗ 𝒙 ∗ 𝒙
𝟗𝒙 = 𝟑 ∗ 𝟑 ∗ 𝒙
In all of them we have 9x
𝟗𝒙 𝟐+𝟓𝒙𝟐
𝟗𝒙 𝟏
𝟐+𝟓𝒙𝟐
𝟏