# Greatest Common Factors: Factoring by Grouping Contributed by: OBJECTIVES:
1. Factor out the greatest common factor. 2. Factor by grouping.
Sec 7.1 - 1
2. Chapter 7
Factoring
Sec 7.1 - 2
3. 7.1
Greatest Common Factors;
Factoring by Grouping
Sec 7.1 - 3
4. 7.1 Greatest Common Factors; Factoring by Grouping
Objectives
1. Factor out the greatest common factor.
2. Factor by grouping.
5. 7.1 Greatest Common Factors; Factoring by Grouping
Factor Out the Greatest Common Factor
Just as we can multiply 2 and 4 to get 8, we can unmultiply 8 to
obtain 2 and 4. We call such unmultiplication factoring.
Similarly, we can sometimes factor a polynomial by writing it
as the product of two or more simpler polynomials.
Both processes use the distributive property. Multiplying
“undoes” factoring, and factoring “undoes” multiplying.
6. 7.1 Greatest Common Factors; Factoring by Grouping
Factoring Out the Greatest Common Factor
The first step in factoring is to find the greatest common
factor (GCF) – the largest term that divides each term of the
polynomial. The GCD is a factor of all the terms of the
7. 7.1 Greatest Common Factors; Factoring by Grouping
Factoring Out the Greatest Common Factor
8. 7.1 Greatest Common Factors; Factoring by Grouping
Factoring Out a Binomial Factor
The greatest common factor need not be a monomial.
Think of this as two
terms with a common
factor of (a + b).
Think of this as two
terms with a common
factor of a(b – c)2.
9. 7.1 Greatest Common Factors; Factoring by Grouping
Factoring Out a Negative Common Factor
When the coefficient of the term of greatest degree is
negative, it is sometimes preferable to factor out the –1 that is
understood along with the GCF.
Either is
correct.
Factor only
the 3 out.
Or factor
the –3 out.
10. 7.1 Greatest Common Factors; Factoring by Grouping
Factoring by Grouping
Many polynomials have no greatest common factor other
than the number 1. Some of these can be factored using the
distributive property if those terms with a common factor are
grouped together. Consider the polynomial:
1. The first two terms have a 5 in common, whereas,
2. The last two terms have an x in common.
Applying the distributive property, we have
11. 7.1 Greatest Common Factors; Factoring by Grouping
Factoring by Grouping
This last expression can be thought of as having two terms,
and .
Applying the distributive property again to factor (x + y)
from each term:
12. 7.1 Greatest Common Factors; Factoring by Grouping
Factoring by Grouping
.