1. 6.1 & 6.2 Greatest Common Factor and Factoring by Grouping
CHAPTER 6
Factoring
6.1 & 6.2 Greatest Common Factor and Factoring by Grouping
Copyright © 2011 Pearson Education, Inc.

2. Greatest Common Factor and Factoring by Grouping
6.1 & 6.2
1. Find the greatest common factor of a set of terms.
2. Factor a monomial GCF out of the terms of a polynomial.
3. Factor polynomials by grouping.
Copyright © 2011 Pearson Education, Inc.

3. Copyright © 2011 Pearson Education, Inc.
Factored form: A number or expression written as a product of factors.
Following are some examples of polynomials in factored form:
2x + 8 = 2(x + 4) x2 + 5x + 6 = (x + 2)(x + 3)
Factored form
Factored form
Greatest common factor (GCF) of a set of terms: A monomial with the greatest coefficient and degree that evenly divides all of the given terms.
Copyright © 2011 Pearson Education, Inc.

4. Copyright © 2011 Pearson Education, Inc.
Finding the GCF
To find the GCFof two or more monomials,
1. Write the prime factorization in exponential form for each monomial. Treat variables like prime factors.
2. Write the GCF’s factorization by including the prime factors (and variables) common to all the factorizations, each raised to its smallest exponent in the factorizations.
3. Multiply the factors in the factorization created in step 2.
Note: If there are no common prime factors, then the GCF is 1.
Copyright © 2011 Pearson Education, Inc.

5. Copyright © 2011 Pearson Education, Inc.
Example 1
Find the GCF of 45a3b and 30a2.
Solution
Write the prime factorization of each monomial, treating the variables like prime factors.
45a3b = 32 • 5 • a3 • b
30a2 = 2 • 3 • 5 • a2
The common prime factors are 3, 5, and a.
GCF = 3 • 5 • a2 = 15a2
Copyright © 2011 Pearson Education, Inc.

6. Copyright © 2011 Pearson Education, Inc.
Factoring a Monomial GCF Out of a Polynomial
To factor a monomial GCF out of the terms of a polynomial,
1. Find the GCF of the terms in the polynomial.
2. Rewrite the polynomial as a product of the GCF and the quotient of the polynomial and the GCF.
Polynomial =
Copyright © 2011 Pearson Education, Inc.

7. Example 2a Solution Find the GCF of 36x2 + 42x. 36x2 + 42x = 6x(6x+7)
Factor. 36x2 + 42x
Solution
Find the GCF of 36x2 + 42x.
Rewrite using the form
36x2 + 42x
Separate the terms.
= 6x(6x+7)
Divide the terms by the GCF.
It is possible to check your answer by distributing.
Copyright © 2011 Pearson Education, Inc.

8. Copyright © 2011 Pearson Education, Inc.
Example 2b
Factor
Solution
The GCF is 5x3.
Copyright © 2011 Pearson Education, Inc.

9. Copyright © 2011 Pearson Education, Inc.
Example 3b
Factor.
Solution
Because the first term is negative, we will factor out the negative of the GCF; which is 3x2 y.
Copyright © 2011 Pearson Education, Inc.

10. Copyright © 2011 Pearson Education, Inc.
Example 4 (GCF)
Factor.
Solution
Notice that this expression is a sum of two products, a(b + 5), and 8(b + 5). Further, note that (b + 5) is the GCF of the two products.
Copyright © 2011 Pearson Education, Inc.

11. Copyright © 2011 Pearson Education, Inc.
Factoring by Grouping
To factor a four-term polynomial by grouping,
1. Factor out any monomial GCF (other than 1) that is common to all four terms.
2. Group together pairs of terms and factor the GCF out of each pair.
3. If there is a common binomial factor, then factor it out.
4. If there is no common binomial factor, then interchange the middle two terms and repeat the process. If there is still no common binomial factor, then the polynomial cannot be factored by grouping.
Copyright © 2011 Pearson Education, Inc.

12. Copyright © 2011 Pearson Education, Inc.
Example 5a
Factor.
Solution
There is no monomial GCF. Because the polynomial has four terms, we try to factor by grouping.
Copyright © 2011 Pearson Education, Inc.

13. Copyright © 2011 Pearson Education, Inc.
Example 5b
Factor.
Solution
There is no monomial GCF. Because the polynomial has four terms, we try to factor by grouping.
Copyright © 2011 Pearson Education, Inc.

14. Copyright © 2011 Pearson Education, Inc.
Example 5c
Factor.
Solution
There is a monomial GCF, 3p, common to all four terms.
Because the polynomial in the parenthesis has four terms, we try to factor by grouping.
Copyright © 2011 Pearson Education, Inc.

15. Copyright © 2011 Pearson Education, Inc.
Factor by factoring out the GCF. a) b) c) d)
Copyright © 2011 Pearson Education, Inc.
6.1

16. Copyright © 2011 Pearson Education, Inc.
Factor by factoring out the GCF. a) b) c) d)
Copyright © 2011 Pearson Education, Inc.
6.1

17. Copyright © 2011 Pearson Education, Inc.
Factor by grouping. a) b) c) d)
Copyright © 2011 Pearson Education, Inc.
6.1

18. Copyright © 2011 Pearson Education, Inc.
Factor by grouping. a) b) c) d)
Copyright © 2011 Pearson Education, Inc.
6.1