section 6.5 factoring by grouping and a general strategy for factoring polynomials

Section 6.5

Factoring by Grouping and a General Strategy for Factoring Polynomials

Objective 1: Factor polynomials by the method of grouping.

Your goal upon finishing this section is to be able to use the various methods covered in this chapter to factor each given polynomial. In this first set of problems it is necessary to group pairs of terms as was done in Section 6.1. It may be necessary to reorder the terms to find a useful grouping.

6.5 Lecture Guide: Factoring by Grouping and a General Strategy for Factoring Polynomials

Completely factor the following polynomials.

1. 3 2x a b a b

Completely factor the following polynomials.

2. 3 2 4 4x x x

3. 6 4 3 4xy x y

Completely factor the following polynomials.

4. 3 22 18 5 45x x x y y

Completely factor the following polynomials.

5.

Factor each polynomial by using the special pattern

. 2 2A B A B A B

2 23 1 25x y

6.

Factor each polynomial by using the special pattern

. 2 2A B A B A B

2249 2x a b

Sometimes it is necessary to group 3 terms together. Start by trying to pick out a perfect square trinomial.

7. 2 236 12 1 49x x y

Sometimes it is necessary to group 3 terms together. Start by trying to pick out a perfect square trinomial.

8. 2 225 6 9x y y

Objective 2: Determine the most appropriate method for factoring a polynomial.

After factoring out the GCF (greatest common factor), proceed as follows. Binomials: Factor special forms:

_________ of Two Squares

Difference of Two_______

_________ of Two Cubes

is prime The sum of two squares is____________ if and

are only second-degree terms and have no common factor other than 1.

2 2A B A B A B

3 3 2 2A B A B A AB B

3 3 2 2A B A B A AB A

2 2A B2A

2B

Strategy for Factoring a Polynomial Over the Integers

After factoring out the GCF (greatest common factor), proceed as follows.

Trinomials: Factor the forms that are perfect squares:

Perfect Square Trinomial; a Square of a Sum

Perfect Square Trinomial; a Square of a Difference

Factor trinomials that are not perfect squares by inspection if possible; otherwise, use the trial-and-error method or the AC method.

Polynomials of Four or More Terms: Factor by grouping

22 22A AB B A B

22 22A AB B A B

9. 2 225 121x y

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

10. 4 2 23 75x x y

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

11.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

3 22 18 36x x x

12.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

4 210 9x x

13.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

2 29 30 25x xy y

14.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

225 100 36x x

15.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

3 31250 80x y

16.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

5 2 46 48x y x y

17.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

2 2 10 10x y x y

18.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

2 260 12 40 8ax y axy x y xy

19.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

3 3 4 4x y x y

20.

Factor each polynomial completely. If it is prime write “Prime” and justify your answer.

3 38 5 10x y x y