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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.

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

5 · 3

Factor Factor

5x2 · 3y4

Factor Factor

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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

2 is also a factor of 12 and 6, but 6 is the greatest common factor.

6 (2 )x y z

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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.

Factoring a Polynomial With Common Factors

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Example

Factor. 36a6 + 45a4 – 18a2

36a6 + 45a4 – 18a2

Check: 9a2(4a4 + 5a2 – 2) = 36a6 + 45a4 – 18a2

2 4 29 4 5 2a a a

9a2 is the common factor.

Multiplying the factors yields the original polynomial.

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Example

Factor.

The GCF is 8x2y

3 2 2 3 28 16 24x y x y x y

3 2 2 3 28 16 24x y x y x y 2 28 2 3x y x y xy

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Example

Factor.

The GCF is 9a2b2

3 2 2 29 9a b a b

3 2 2 29 9a b a b 2 29 1a b a

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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 )(6 )x y z

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Example

Factor.

The common factor is 2x – 3y.

27 (2 3 ) (1 2 3 )x x y x y

27 (2 3 ) (2 3 )x x y x y

27 (2 3 ) (2 3 )x x y x y

2(2 3 )(7 1)x y x