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Section 6.1
Removing a
Common Factor
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Common Factors
When two or more numbers, variables, or algebraic
expressions are multiplied, each is called a factor.
5x2 · 3y4
5·3
Factor
Factor
Factor
Factor
When asked to factor a number or algebraic expression,
you are being asked to determine what factors, when
multiplied, will result in that number or expression.
Factor. 15xy = 3 · 5 · x · y
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Example
Factor. 15x – 5
15x – 5 = 5(x – 3)
Find a factor both terms
have in common.
Rewrite the expression
as a product.
Check:
5(x – 3) = 15x – 5
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Example
Factor. 12xy + 6xz
12xy + 6xz  6 x(2y  z )
2 is also a factor of 12 and 6, but 6 is the greatest
common factor.
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Factoring a Polynomial With Common
Factors
1.
Determine the greatest numerical common factor by
asking, “What is the largest integer that will divide
into the coefficient of all the terms?”
2.
Determine the greatest common variable factor by
asking, “What variables are common to all the
terms?” Then, for each variable that is common to all
the terms, ask, “What is the largest exponent of the
variable that is common to all the terms?”
3.
The common factors found in steps 1 and 2 are the
first part of the answer.
4.
After removing the common factors, what remains is
placed in parentheses as the second factor.
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Example
Factor. 36a6 + 45a4 – 18a2

36a6 + 45a4 – 18a2  9a 2 4a 4  5a 2  2

9a2 is the common factor.
Check: 9a2(4a4 + 5a2 – 2) = 36a6 + 45a4 – 18a2 
Multiplying the factors yields the original
polynomial.
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Example
Factor. 8 x 3 y  16 x 2 y 2  24 x 3 y 2
The GCF is 8x2y

8 x 3 y  16 x 2 y 2  24 x 3 y 2  8 x 2 y x  2y  3 xy 2
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
Example
Factor. 9a3b 2  9a 2b 2
The GCF is 9a2b2
2 2

9
a
b  a  1
9a b  9a b
3
2
2
2
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Example
Factor. 6(3x + y) – z(3x + y)
The common factor is 3x + y.
6(3x + y) – z(3x + y)  (3 x  y )(6  z )
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Example
2
Factor. 7 x (2x  3y )  (2 x  3 y )
The common factor is 2x – 3y.
7 x 2 (2x  3y )  (2 x  3 y )  7 x 2 (2x  3y )  1(2x  3y )
 (2x  3y )(7 x 2  1)
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