Section 6.2 Factoring Trinomials of the Form x 2 + bxy + cy 2.

Section 6.2

Factoring Trinomials of the Form

x2 + bxy + cy2

6.2 Lecture Guide: Factoring Trinomials of the Form x2 + bxy + cy2

Objective: Factor trinomials of the form 2x bx c

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

Your goal for this section should be to be able to factor trinomials as quickly and efficiently as possible. There are two new methods presented in this section for factoring trinomials: 1) Factoring by Tables and Grouping and 2) Factoring by Inspection. Most trinomials you will see should be factored by inspection. The method of tables and grouping is presented first to illustrate the connection between factoring and multiplication.

Factorable Trinomials:

Algebraically

Verbally

where

21 2x bx c x c x c

1 2c c c and 1 2c c b

A trinomial is factorable into a pair of binomial factors with integer coefficients if and only if there are two integers whose ____________ is c and whose ____________ is b. Otherwise, the trinomial is prime over the integers.

2x bx c

Example 2 14 24 2 12x x x x

where and2 12 ______ 2 12 ______

Factoring Trinomials of the Form by Tables and

Grouping:

2x bx c

Example 2 10 24x x Factor

Step 2: Find a pair of factors of c whose sum is b. If there is not a pair of factors whose sum is b, the trinomial is prime over the integers.

• If the constant c is positive, the factors of c must have the same sign. These factors will share the same sign as the linear coefficient, b.

• If the constant c is negative, the factors of c must be opposite in sign. The sign of b will determine the sign of the factor with the larger absolute value.

ProcedureStep 1: Be sure you have factored out the GCF if it is not 1.

Factors of –24 Sum of Factors

–1

–2

–3

– 4

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 so that b is the sum of the pair of factors from Step 2.

2

2

10 24

______ ______ 24

12

12

x x

x

x

x

Factors of –24 Sum of Factors

–1

–2

–3

– 4

Factoring Trinomials of the Form by Tables and

Grouping:

2x bx c

Factors of _____ Sum of Factors

1.

Factor each polynomial using the method of tables and grouping.

2 14 24x x

Multiply the factors to check your work.


2.

Factor each polynomial using the method of tables and grouping.

Multiply the factors to check your work.

2 10 56x x


3.

The method of tables and grouping also provides an organized way to show that a trinomial is prime.

This trinomial is prime because no two factors of ______ have a sum of ______.

2 14 36x x

Factoring by InspectionThe method of factoring by tables and grouping is especially useful when the coefficients are relatively large and for verifying that you have checked all possibilities before declaring that a polynomial is prime. This method will also work for small coefficients. However, you should work on factoring polynomials that have relatively small coefficients by inspection. To factor 2x bx c by inspection, try to mentallydetermine two factors of c whose sum is b.

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

2 21 20x x = x x 4.

2 9 20x x x x = 5.


= 6.

= 7.

2 11 30x x x x

2 13 30x x x x


= 8.

= 9.

2 4 12x x x x

2 12x x x x


= 10.

= 11.

2 16 36x x x x

2 5 36x x x x

Factor each trinomial. If it is prime, write "Prime" and justify your result.

12. 2 8 12x x


13. 2 8 15x x


14. 2 4 3x x


15. 2 8 9x x


16. 2 10 24t t


17. 2 10 21m m


18. 2 8 7y y


19. 2 16 15y y


20. 2 14 15a a


21. 2 8 9t t


22. 2 12x x


23. 2 6 16x x


24. 2 2b b


25. 2 3 28z z


26. 2 5 28x x


27. 2 20n n


28. 2 12 20c c


29. 2 3 40x x

Factor each trinomial.

30. 2 222 48x xy y


31. 2 25 36x xy y


32. 2 211 24x xy y


33. 2 28 12x xy y


34. 2 214 48x xy y


35. 2 211 10x xy y

Each of the following trinomials has a greatest common factor. First factor out the GCF and then complete the factorization.

36. 22 2 12x x


37. 25 45 100x x


38. 210 20 30x x


39. 2 4 12x x


40. 2 9 18x x


41. 23 15 42x x


42. 22 12 10x x


43. 3 2 2 33 9 30x y x y xy


44. 3 3 2 4 52 8 90x y x y xy

METHODS FOR FACTORING 2x bx c

ADVANTAGES

DISADVANTAGES

Graphs

Visually displays the x-intercepts that correspond to the factors of the trinomial. If there are no x-intercepts, this indicates the polynomial is prime over the integers.

Can be time-consuming to select the appropriate viewing window and to approximate the x-intercepts. If the x-intercepts are near integer values but not exactly on these values, it is easy to be mislead by the graph and to select incorrect factors.


Tables

ADVANTAGES

DISADVANTAGES

Can see the zeros of the trinomial and can observe numerical patterns that are important in many applications. Spreadsheets allow us to use the power of computers to exploit this method.

Often requires insight or some trial and error in order to select the most appropriate table.


Tables and Grouping

ADVANTAGES

DISADVANTAGES

This is a precise step-by-step process that can factor any trinomial of the form

Many trinomials with small integer coefficients can be factored by inspection and it is not necessary to write the table and all the steps of this method.

2x bx c or can identify the trinomialas prime. This method has the same steps used to multiply binomial factors only the steps are reversed.


Inspection

ADVANTAGES

DISADVANTAGES

Takes advantage of patterns and insights to quickly factor trinomials with small integer coefficients. Observing the mathematical patterns in these trinomials can improve your foundation for other topics.

This can become a trial-and-error process that requires considerable insight. For novices this process can be very challenging. It is important to examine all possibilities before deciding the trinomial is prime.

Section 6.2 Factoring Trinomials of the Form x 2 + bxy + cy 2.

Documents

Transcript of Section 6.2 Factoring Trinomials of the Form x 2 + bxy + cy 2.