Section 6.2 Factoring Trinomials of the Form x 2 + bx + c.

Section 6.2

Factoring Trinomials of the Form

x2 + bx + c

6.2 Lecture Guide: Factoring trinomials of the form x2 + bx + c

Objective 1: Factor trinomials of the form by trial-and-error method or by inspection.

2x bx c

A trinomial in the form is called a quadratic trinomial. Your goal for this section should be to beable to factor quadratic trinomials as quickly and efficiently as possible. We will restrict our focusto the case where the leading coefficient a = 1 for this section.

2ax bx c

To begin understanding the procedure we use to factor these trinomials, examine the following three products.

Factors F O I L Product

= +12 =

= +12 =

= +12 =

1 12x x 12x x 2 13 12x x 2x

6x 2x2 8 12x x

2x

2 7 12x x

2 6x x

3 4x x 2x 4x 3x

Sum is the linear term

Product is 1 Product is 12

Thus, a factorization of must have a pair of factors of c whose sum is b.

2x bx c

Factorable Trinomials

Algebraically Verbally Example

Where

And .

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.

Where

and .

is prime because there are no factors of ______ whose sum is ______.

2

1 2

x bx c

x c x c

1 2c c c1 2c c b

2x bx c

2 14 24

2 12

x x

x x

2 12 ____ 2 12 ____

2 50 24x x

To factor by trial and error list the possible factors of c and examine them until you find the right combination that produces the correct linear term. To factor by inspection, try to mentally determine two factors of c whose sum is b. Initially, it is a good idea to list the factors. This will help you to notice patterns and develop a sense for factoring trinomials. With practice you will be able to factor many of the trinomials in this section by inspection.

Factoring by Trial-and-Error or by Inspection2x bx c

2x bx c

2 21 20x x

x x

2 9 20x x

x x

1 20

2 10

4 5

1 20

2 10

4 5

1. 2.

In problems 1-8, the values below the binomial factors represent the possible factors of the constant term c in the quadratic trinomial. Use this information along with the given sign pattern to determine the factored form.

1 30

2 15

3 10

5 6

1 30

2 15

3 10

5 6

3. 4.2 11 30x x 2 13 30x x

x x x x

1 12

2 6

3 4

1 12

2 6

3 4

5. 6.2 4 12x x 2 12x x

x x x x

1 36

2 18

3 12

4 9

6 6

7. 8.2 16 36x x 2 5 36x x

x x x x

1 36

2 18

3 12

4 9

6 6

Note: What is wrong with problem 8? How can you fix it?

Verbally Algebraically Examples

If the constant term is ____________, the factors of this term must have the same sign. These factors will share the same sign as the linear term.If the constant term is ____________, the factors of this term must be opposite in sign. The sign of the constant factor with the larger absolute value will be the same as that of the linear term.

2

? ?

x bx c

x x

2

? ?

x bx c

x x

2 5 6

2 3

x x

x x

2 5 6

2 3

x x

x x

2

? ?

x bx c

x x

2

? ?

x bx c

x x

2 5 6

6 1

x x

x x

2 5 6

1 6

x x

x x

2 5 6x x

2 5 6x x

2 5 6x x

2 5 6x x

___ ___x x

___ ___x x

___ ___x x

Match each trinomial with the appropriate sign pattern for the factors of this trinomial.

9.

10.

11.

12.

A.

B.

C.

Factor each trinomial by trial-and-error or by inspection.

13. 14.2 8 12x x 2 8 15x x


15. 16.2 10 21m m 2 10 24t t


17. 18.2 8 7y y 2 14 15a a


19. 20.2 12x x 2 6 16x x


21. 22.2 3 28z z 2 12 28x x


23. 24.2 12 20c c 2 3 40x x

Objective 2: Identify a prime trinomial of the form .

2x bx c

A Prime Polynomial

A polynomial is prime over the integers if its only factorization must involve either ____ or _____ as one of the factors.

1 36

2 18

3 12

4 9

6 6

Factors of 36

Use the given tables to assist you in identifying which trinomials are prime and factoring the trinomials that can be factored over the integers. Note that if we list all possibilities in a table and none of them work, then the polynomial must be prime.

25. 2 15 36x x

37

20

15

13

12

Sum of Factors


26. 2 14 36x x

Sum of FactorsFactors of 36

1 36

2 18

3 12

4 9

6 6

37

20

15

13

12

Factors of 24


27. 2 14 24x x

Sum of Factors

−1 −24

−2 −12

−3 −8

−4 −6

−25

−14

−11

−10


28.2 13 24x x

Factors of 24 Sum of Factors

−1 −24

−2 −12

−3 −8

−4 −6

−25

−14

−11

−10

Factors of 56


29. 2 20 56x x

Sum of Factors

−1 56

−2 28

−4 14

−7 8

55

26

10

1


30. 2 10 56x x

Factors of 56 Sum of Factors

−1 56

−2 28

−4 14

−7 8

55

26

10

1

Objective 3: Factor trinomials of the form by the trial-and-error or by inspection.

2 2x bxy cy


31.2 222 48x xy y


32.2 25 36x xy y


33.2 211 24x xy y


34.2 28 12x xy y


35.2 214 48x xy y


36.2 211 10x xy y

22 2 12x x

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

37.

25 45 100x x


38.


39. 210 20 30x x


40. 2 4 12x x


41. 2 9 18x x


42. 23 15 42x x

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

22 12 10x x 43.


44.3 2 2 33 9 30x y x y xy


45. 3 3 2 4 52 8 90x y x y xy


46. 22 5 2 36 2a b x a b x a b

Polynomial Factored Form Zeros of

x-intercepts of the graph of

P x P x y P x2 3 18x x 2 10 21x x

47.

48.

Complete the table for each polynomial.

Section 6.2 Factoring Trinomials of the Form x 2 + bx + c.

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Transcript of Section 6.2 Factoring Trinomials of the Form x 2 + bx + c.