Math 90
5.1 "Factoring Out the Greatest Common Factor and
Factoring by Grouping"
Bibiana Lopez
Crafton Hills College
April 2010
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Objectives:
* Find the greatest common factor of a list of terms.
* Factor out the greatest common factor.
* Factor by grouping.
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Factoring Monomials
De…nition
The Greatest Common Factor (GCF):
The greatest common factor of a list of integers is the largest common
factor of those integers.
Fact
Finding the Greatest Common Factor (GCF):
1. Find the prime factorization of each monomial.
2. List each common factor the least number of times it appears in any
one monomial.
3. Find the product of the factor found in the list to obtain the GCF.
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Factoring Monomials
Example 1: (GCF)
Find the GCF of:
a) 24x 2 y 3 , 15x 3 y , and 18x 2 y 2
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b) 20m2 w 3 , 12mw 4 , and 8m3 w 2
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Factor Out the Greatest Common Factor
Example 2: (Factoring out the greatest common factor)
Factor the following polynomials.
a) 9x 4 y 2 12x 3 y 3
b) 3t 2 + 15t
c) 4a4 b 2 + 6a3 b 2 + 2a2 b
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d) 6xy 2 z
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12x 2 yz + 3xy
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Factoring Out a Negative Factor
It is often useful to factor out the negative of a polynomial.
Example 3: (Factoring out a negative factor)
Factor the following polynomials.
a)
18x 5 + 6x 3 12x 2
b)
25xy 2
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15x 2 y + 30x 2 y 2
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Factoring Out a Polynomial
Example 4: (Factoring out a polynomial)
Factor the following polynomials.
a) 2 (x + 3)2 + 4 (x + 3)
b) 4p 3 q 2 (2a + b ) + 8p 2 q 3 (2a + b )
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Factor by Grouping
Fact
Factoring by Grouping:
1. Group the terms of the polynomial so that each group has a common
factor.
2. Factor out the common factor from each group.
3. Factor out the resulting common factor.
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Factor by Grouping
Example 5: (Factor by grouping)
Factor by grouping.
a) x 5 + 5x 4 + x + 5
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b) y 3
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6 + 3y 2
2y
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Factor by Grouping
c) 2tx + 3dy + ty + 6dx
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d) 3x 3 y
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4x 2 y 2
6x 2 y + 8xy 2
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