Beginning Algebra, 11th Edition

CHAPTER 6 Summary^411

6.1 The Greatest Common Factor; Factoring

by Grouping

Finding the Greatest Common Factor (GCF)

Step 1 Write each number in prime factored form.

Step 2 List each prime number or each variable that is a
factor of every term in the list.

Step 3 Use as exponents on the common prime factors
the leastexponents from the prime factored forms.

Step 4 Multiply the primes from Step 3.

Find the greatest common factor of and

The greatest common factor is 2xy.

2 xy^2 = 2 #x#y^2

6 x^2 y^3 = 2 # 3 #x^2 #y^3

4 x^2 y= 2 # 2 #x^2 #y

4 x^2 y, 6 x^2 y^3 , 2 xy^2.

QUICK REVIEW

CONCEPTS EXAMPLES

Factoring by Grouping

Step 1 Group the terms.

Step 2 Factor out the greatest common factor in each
group.

Step 3 Factor out a common binomial factor from the
results of Step 2.

Step 4 If necessary, try a different grouping.

Factor by grouping.

Group the terms.

Factor each group.

= 13 x+ 521 x- 8 y 2 Factor out 3x+5.

=x 13 x+ 52 - 8 y 13 x+ 52

= 13 x^2 + 5 x 2 + 1 - 24 xy- 40 y 2

3 x^2 + 5 x- 24 xy- 40 y

6.2 Factoring Trinomials

To factor find mand nsuch that
and

Then factors as

Check by multiplying.

x^2 +bx+c 1 x+m 21 x+n 2.

m+n=b

x^2 +bx+c

mn=c

m+n=b.

x^2 +bx+c, mn=c Factor

and

CHECK

=x^2 + 6 x+ 8 ✓

=x^2 + 4 x+ 2 x+ 8

1 x+ 221 x+ 42

x^2 + 6 x+ 8 factors as 1 x+ 221 x+ 42.

m+n= 6

x^2 + 6 x+ 8 m= 2 n= 4

mn= 8

x^2 + 6 x+ 8.

6.3 More on Factoring Trinomials

To factor use one of the following methods.

Grouping
Find mand n.

Trial and Error
Use FOIL in reverse.

m+n=b

ax^2 +bx+c

mn=ac

ax^2 +bx+c,

Factor.

The required integers are.

By trial and error or by grouping,

3 x^2 + 14 x- 5 factors as 13 x- 121 x+ 52.

m=-1 and n= 15

mn=- 15 , m+n= 14

• 15

3 x^2 + 14 x- 5

(continued)