Section 6.3 Factoring Trinomials of the Form ax 2 + bx + c.

Section 6.3

Factoring Trinomials of the Form

ax2 + bx + c

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

2ax bx c

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

1.22 7 6x x

2x x


2.22 13 6x x

2x x


3.23 11 6x x

3x x


4.23 19 6x x

3x x


5. 23 4x x

3x x


6. 23 4 4x x

3x x


7. 25 17 12x x

5x x


8. 25 4 12x x

5x x

Fill in the missing + or – symbols to complete the factorization of each trinomial.

22 7 5x x

2 ____5 ____1x x

9.


23 8 4x x

3 ____ 2 ____ 2x x

10.


11. 24 8 5x x

2 ____1 2 ____5x x


12. 25 7 6x x

5 ____3 ____ 2x 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.

13. 14.22 7 3x x 22 11 5x x


15. 16.22 3x x 22 9 5x x


17. 18.22 5 3x x 22 3 5x x

2ax bx c Objective 2: Factor trinomials of the formby the AC method.

Objective 3: Identify a prime trinomial of the form .

2ax bx c

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 theleading 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.

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

19.

FactorsProducts

F O I L

1 6 15x x 26 15 6 15x x x 26 21 15x x

3 6 5x x 26 15x

5 6 3x x

15 6 1x x

2 1 3 15x x

2 3 3 5x x 2 5 3 3x x

2 15 3 1x x

26 15x 26 15x 26 15x 26 15x 26 15x 26 15x

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?

20.


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

(d) What is the correct factorization of ?

20.

26 19 15x x


(e) The procedure for factoring trinomials of the form by the AC method involves finding two

factors of ac whose sum is b. When expanded, the correct factorization of has two middle terms whose coefficients have a product of ____________ and a sum of ____________.

20.

26 19 15x x

2ax bx c

Example

Factor

ProcedureStep 1: Factor out the GCF. If a is negative, factor out -1.

Factors of –120

–1

–2

–3

–4

–5

–6

–8

–10

Sum of Factors

26 7 20x x Step 2: Find a pair of factors of ac whose sum is b. If there is nota pair of factors whose sum is b, the trinomial is prime over the integers.If the constant c is positive, the factors of ac must have the _________ _________. These factors will share the same sign as the linear coefficient b.If the constant c is negative, the factors of ac must be _________ in sign. The sign of b will determine the sign of the factor with the larger absolute value.

Factoring by the AC Method2ax bx c

Factoring by the AC Method2ax bx c

Step 3: Rewrite the linear term of so that b is the sum of the factors from Step 2.

2ax bx c

Step 4: Find a pair of factors of ac whose sum is b. If there is nota pair of factors whose sum is b, the trinomial is prime over the integers.

Example

Factor Factors of –120

–1

–2

–3

–4

–5

–6

–8

–10

Sum of Factors

26 7 20x x

2

2

6 7 20

6 ______ ______ 20

2 5

2 5

x x

x

x

x

21.Factor using theAC method.

Factors of Sum of ______ Factors

Multiply the factors to check your work.

22 9 4x x




22 11 14x x




24 17 15x x



Multiply the factors to check your work.210 11 6x x

Factor each trinomial by inspection or by the AC method. If itIs prime, write "Prime" and justify your result.

25. 25 29 20x x


26. 24 19 12x x


27. 26 35y y


28. 26 13 5x x


29. 210 19 6x x


30. 28 35 12x x


31. 2 28 26 45x xy y


32. 2 212 7 10x xy y

First factor out −1.

33. 22 5 12x x

First factor out −1.

34. 23 7 6x x

First factor out the GCF.

35. 24 20 56x x


36. 220 70 40x x


37. 3 210 25 15x x x


38. 3 26 57 105x x x


39. 3 2 2 3 436 66 80x y x y xy


40. 3 2 2 324 102 45a b a b ab

METHODS FOR FACTORING

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. Because the x-intercepts are approximated, these factors should be checked.

2ax bx c



Tables 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.

2ax bx c



AC Method This is a precise step-by-step process that can factor any trinomial of the form or can identify the trinomial as prime. This method has the same steps used to multiply binomial factors but the steps are reversed.

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.

2ax bx c

2ax bx c



Trial-and-Error Method

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.

For novices this process can be very challenging. It is important to examine all possibilities before deciding the trinomial is prime.

2ax bx c

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Section 6.3 Factoring Trinomials of the Form ax 2 + bx + c.

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