Section 6.3

Factoring Trinomials of the Form

ax2 + bxy + cy2

7.2 Lecture Guide: Factoring Trinomials of the Form ax2 + bxy + cy2

Objective 1: Factor trinomials of the form 2ax bx c

Objective 2: Factor trinomials of the form 2 2ax bxy cy

Factoring a polynomial can be considered a reversal of the process of multiplying the factors of the polynomial. In Section 6.2, we focused on factoring trinomials where the leading coefficient was 1. Factoring trinomials where the leading coefficient is not 1 can be more complicated. We will start by multiplying several pairs of factors that form a trinomial with a leading coefficient of 6.

Factors FIRST MIDDLE LAST Products

1 6 15x x 26 15 6 15x x x

1. Multiply the factors in this table by writing out both middle terms and then simplify the result. The first row has been completed.

3 6 5x x 26 15x

5 6 3x x

15 6 1x x

26 15x

26 15x

Factors FIRST MIDDLE LAST Products

26 15x

26 15x

26 15x

26 15x 2 1 3 15x x

2 3 3 5x x

2 5 3 3x x

2 15 3 1x x

1. Multiply the factors in this table by writing out both middle terms and then simplify the result. The first row has been completed.

2. Answer each question about the table above. (a) What is the product of the coefficients of each pair of middle terms?

(b) What do you notice about the first and last term of each product?

(c) What is the product of the coefficients of the first and last terms?

(d) What is the correct factorization of 26 19 15x x ?

(e) The procedure for factoring trinomials of the form 2ax bx c by tables and grouping involves finding two

factors of ac whose sum is b. When expanded, the correctfactorization of 26 19 15x x has two middle terms whosecoefficients have a product of ____________ and a sumof ____________.

Example

Factor ProcedureStep 1: Be sure you have factored out the GCF if it is not 1. (If a<0, factor out -1.)

Factors of –120

–1

–2

–3

–4

–5

–6

–8

–10

Sum of Factors

26 7 20x x

Step 2: Find two factors of ac whose sum is b.

Factoring 2ax bx c by Tables and Grouping

Note: If ac is positive then both factors must have the __________ __________.

If ac is negative, then the two factors will have the ___________ sign with the “larger” factor having the same sign as b.

2

2

6 7 20

6 ______ ______ 20

2 5

2 5

x x

x

x

x

Example

Step 4: Factor the polynomial from Step 3 by grouping the terms and factoring the GCF out of each pair of terms.

ProcedureStep 3: Rewrite the linear term of ax2 + bx + c so that b is the sum of the pair of factors from Step 2.

Factors of –120

–8

Sum of Factors

Factoring 2ax bx c by Tables and Grouping

3. 22 5 12x x

Factor each polynomial using the method of tables and grouping.

Factors of ___ Sum of Factors

Multiply the factors to check your work.

4.

Factor each polynomial using the method of tables and grouping.

Factors of ___ Sum of Factors

Multiply the factors to check your work.

22 9 4x x

5.

Factor each polynomial using the method of tables and grouping.

Factors of ___ Sum of Factors

Multiply the factors to check your work.

210 11 6x x

Fill in the missing information to complete the factorization of each trinomial by inspection.

6.

8. 9.

7.22 7 6x x

2x x

22 13 6x x

2x x

23 11 6x x 3x x

23 19 6x x 3x x

Fill in the missing information to complete the factorization of each trinomial by inspection.

12.

10. 11.

13.

23 4x x

3x x

23 4 4x x

3x x

25 17 12x x

5x x

25 4 12x x

5x x

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection.

14. 22 5 3x x

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection.

15. 22 3 5x x

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection.

16. 22 3x x

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection.

17. 22 9 5x x

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection.

18. 22 7 3x x

It is very useful to be able to recognize patterns and be able to use these patterns to quickly factor some polynomials by inspection. Factor each trinomial by inspection.

19. 22 11 5x x

Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result.

20. 26 35y y

Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result.

21. 26 13 5x x

Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result.

22. 210 19 6x x

Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result.

23. 28 35 12x x

Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result.

24. 2 28 26 45x xy y

Factor each trinomial by inspection or by tables and grouping. If it is prime, write "Prime" and justify your result.

25. 2 212 7 10x xy y

Remember to first factor out the GCF.

26. 22 5 12x x

Remember to first factor out the GCF.

27. 23 7 6x x

Remember to first factor out the GCF.

28. 24 20 56x x

Remember to first factor out the GCF.

29. 220 70 40x x

Remember to first factor out the GCF.

30. 3 210 25 15x x x

Remember to first factor out the GCF.

31. 3 26 57 105x x x

Remember to first factor out the GCF.

32.3 2 2 3 436 66 80x y x y xy

Remember to first factor out the GCF.

33. 3 2 2 324 102 45a b a b ab