Intermediate Algebra (11th edition)

CHAPTER 6 Summary 355

6.1 Greatest Common Factors and

Factoring by Grouping

The Greatest Common Factor
The product of the largest common numerical factor and
each common variable raised to the least exponent that
appears on that variable in any term is the greatest
common factor of the terms of the polynomial.

Factor

= 2 xy 12 x- 25 y 2 The greatest common factor is 2xy.

4 x^2 y- 50 xy^2

4 x^2 y- 50 xy^2.

QUICK REVIEW

CONCEPTS EXAMPLES

Factoring by Grouping

Step 1 Group the terms so that each group has a
common factor.

Step 2 Factor out the common factor in each group.

Step 3 If the groups now have a common factor, factor
it out. If not, try a different grouping.

Always check the factored form by multiplying.

Factor by grouping.

Group the terms.

Factor out 5 and.

= 1 a-b 215 - x 2 Factor out a-b.

= 51 a-b 2 - x 1 a-b 2 - x

= 15 a- 5 b 2 + 1 - ax+bx 2

5 a- 5 b-ax+bx

6.2 Factoring Trinomials

To factor a trinomial, choose factors of the first term and
factors of the last term. Then place them within a pair of
parentheses of this form.

Try various combinations of the factors until the correct
middle term of the trinomial is found.

1212

Factor

The factors of 15 are 5 and 3, and 15 and 1.

The factors of are and 2, 4 and and 8, and 1 and

Various combinations lead to the correct factorization.

= 15 x- 2213 x+ 42 Check by multiplying.

15 x^2 + 14 x- 8

- 8 - 4 - 2,- 1 - 8.

15 x^2 + 14 x-8.

6.3 Special Factoring

Difference of Squares

Perfect Square Trinomials

Difference of Cubes

Sum of Cubes

x^3 y^3  1 xy 21 x^2 xyy^22

x^3 y^3  1 xy 21 x^2 xyy^22

x^2  2 xyy^2  1 xy 22

x^2  2 xyy^2  1 xy 22

x^2 y^2  1 xy 21 xy 2

Factor.

= 14 z+ 12116 z^2 - 4 z+ 12

64 z^3 + 1

= 12 - 3 a 214 + 6 a+ 9 a^22

8 - 27 a^3

= 13 y+ 122 = 14 p- 722

9 y^2 + 6 y+ 1 16 p^2 - 56 p+ 49

= 12 m+ 5 n 212 m- 5 n 2

= 12 m 22 - 15 n 22

4 m^2 - 25 n^2

(continued)