Introduction to Factoring Introduction to Factoring PolynomialsPolynomials

Section 10.4

Finding the Greatest Common FactorFinding the Greatest Common Factor

The first step in factoring a polynomial is to see whether the terms of the polynomial have a common factor.

If there is a common factor, we can write the polynomial as a product by factoring factoring outout the common factor.

We will usually factor out the greatest common factor (GCF).

2Martin-Gay, Prealgebra, 5ed

Finding the Greatest Common Factor. . .Finding the Greatest Common Factor. . .

The greatest common factor (GCF) of a list greatest common factor (GCF) of a list of termsof terms is the product of the GCF of the numerical coefficients and the GCF of the variable factors.

Consider the terms, and .15 2 2x y 6 3x y

15 3 52 2x y x x y y

6 2 33x y x x x y

Common factors denoted by

3 x x yGCF 3 2x y3

Helpful Hint

Notice below that the GCF of a list of terms contains the smallest exponent on each common variable.

The GCF of , , and is x y5 6 x y2 7 x y3 4

x y2 4Smallest exponent on x.

Smallest exponent on y.

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Factoring Out the Greatest Common FactorFactoring Out the Greatest Common Factor

• Do the terms have a greatest common factor other than 1?

To factor a polynomial:

• If so, factor out the greatest common factor from each term by writing each term as a product of the greatest common factor and the term’s remaining factors.

• Use the distributive property to write the factored form of the polynomial.

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Factoring Out the Greatest Common Factor Factoring Out the Greatest Common Factor

• The GCF of 5x + 10 is 5.

Consider, 5x + 10

• Factor 5 from each term and write each term as a product of 5 and the remaining terms,

• Using the distributive property, write

5x + 10 5 5 2x

5 2( )xfactored form of polynomialfactored form of polynomial

Factoring canbe checked by

multiplying.

Factoring canbe checked by

multiplying.

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Helpful Hint

A factored form of 5x + 10 is not

5 5 2 x

Although the termsterms have been factored (written as a product), the polynomialpolynomial 5x + 10 has not been factored. A factored form of 5x + 10 is the productproduct 5(x + 2).

5 5 2 x

5(x + 2)

factored termsfactored terms

factored polynomialfactored polynomial7

Examples of Factored PolynomialsExamples of Factored Polynomials

x x3 5 x x3 21e jDon’t forget the 1.Don’t forget the 1.

8 10 24 2a a a 2 4 5 13a a a e jDon’t forget the -1.Don’t forget the -1.

8 10 24 2a a a 2 4 5 13a a ae jNotice the changes in signs when factoring -2aNotice the changes in signs when factoring -2a

In this example,factor out -2arather than 2a

In this example,factor out -2arather than 2a

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