1. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Section 5.4
Factoring
Trinomials
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

3. Factoring Trinomials
In the previous section we factored certain polynomials having four terms using the method of grouping.
Now, we will use trial and error and a problem solving process to factor trinomials.
The polynomials we will begin with will have leading coefficients of one. We will begin by trying to factor these
trinomials into a product of two binomials. Polynomials
that cannot be factored over a given number set are said to be prime.

4. A Strategy for Factoring T
Factoring Trinomials
A Strategy for Factoring T
1) Enter x as the first term of each factor.
2) List pairs of factors of the constant c.
3) Try various combinations of these factors. Select the combination in which the sum of the Outside and Inside products is equal to bx.
4) Check your work by multiplying the factors using the FOIL method. You should obtain the original trinomial.
If none of the possible combinations yield an Outside product and an Inside product who sum is equal to bx, the trinomial cannot be factored using integers and is called prime over the set of integers. I
Sum of O + I O

5. Factoring Trinomials Factor:
EXAMPLE
Factor:
SOLUTION
1) Enter x as the first term of each factor.
To find the second term of each factor, we must find two integers whose product is 24 and whose sum is 11.
2) List pairs of factors of the constant, 24.
Factors of 24
24, 1
–24, –1
12, 2
–12, –2
8, 3
–8, –3
6, 4
–6, –4

6. Factoring Trinomials
CONTINUED
3) Try various combinations of these factors. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to 11x. Here is a list of the possible factorizations.
Possible Factorizations of
Sum of Outside and Inside Products (Should Equal 11x)
(x + 24)(x + 1)
x + 24x = 25x
(x - 24)(x - 1)
-x - 24x = -25x
(x + 12)(x + 2)
2x + 12x = 14x
(x - 12)(x - 2)
-2x - 12x = -14x
(x + 8)(x + 3)
3x + 8x = 11x
(x - 8)(x - 3)
-3x - 8x = -11x
(x + 6)(x + 4)
4x + 6x = 10x
(x - 6)(x - 4)
-4x - 6x = 10x
This is the required middle term.

7. Factoring Trinomials Thus,
CONTINUED
Thus,
Check this result by multiplying the right side using the FOIL method. You should obtain the original trinomial. Because of the commutative property, the factorization can also be expressed as

8. Factoring Trinomials Factor:
EXAMPLE
Factor:
SOLUTION
The GCF of the three terms of the polynomial is 4y. Therefore, we begin by factoring out 4y. Then we factor the remaining trinomial.
Factor out the GCF
Thus,

9. Objective #1: Example

10. Objective #1: Example

11. Objective #1: Example

12. Objective #1: Example

13. Objective #1: Example

14. Objective #1: Example

15. Objective #1: Example

16. Objective #1: Example

18. Factoring Trinomials using Substitution
In some trinomials, the highest power is greater than 2, and the exponent in one of the terms is half of the other term. By letting u equal the variable to the smaller power, the trinomial can be written in a form that makes its possible factorization more obvious.
If a factorization is found, we replace all occurrences of u in the factorization with the original substitution.

19. Objective #2: Example

20. Objective #2: Example

22. A Strategy for Factoring T
Factoring Trinomials
A Strategy for Factoring T
Assume, for the moment, that there is no greatest common factor.
1) Find two First terms whose product is
2) Find two Last terms whose product is c:
3) By trial and error, perform steps 1 and 2 until the sum of the Outside product and Inside product is bx: I O
Sum of O + I
If no such combinations exist, the polynomial is prime.

23. Factoring Trinomials Factor:
EXAMPLE
Factor:
SOLUTION
1) Find two First terms whose product is There is more than one choice for our two First terms. Those choices are cataloged below. ? ?
2) Find two Last terms whose product is 15. There is more than one choice for our two Last terms. Those choices are cataloged below. ? ?

24. Factoring Trinomials
CONTINUED
3) Try various combinations of these factors. The correct factorization of is the one in which the sum of the Outside and Inside products is equal to 19x. Here is a list of some of the possible factorizations.
Possible Factorizations of
Sum of Outside & Inside Products (Should Equal 19x)
(6x + 1)(x + 15)
90x + x = 91x
(x + 1)(6x + 15)
15x + 6x = 21x
(3x + 3)(2x + 5)
(2x + 3)(3x + 5)
10x + 9x = 19x
(6x + 3)(x + 5)
30x + 3x = 33x
(x + 3)(6x + 5)
5x + 18x = 23x
This is the required middle term.

25. Factoring Trinomials Therefore, the factorization of is:
CONTINUED
Therefore, the factorization of is:
(2x + 3)(3x + 5) .

26. Factoring Trinomials Factor:
EXAMPLE
Factor:
SOLUTION
The GCF of the three terms of the polynomial is We begin by factoring out .
1) Find two First terms whose product is ? ?

27. Factoring Trinomials
CONTINUED

28. Factoring Trinomials CONTINUED (10y - 1)(y - 3) -30y - y = -31y
Possible Factorizations of
Sum of Outside & Inside Products (Should Equal -17y)
(10y - 1)(y - 3)
-30y - y = -31y
(y - 1)(10y - 3)
-3y - 10y = -13y
(2y - 1)(5y - 3)
-6y - 5y = -11y
(5y - 1)(2y - 3)
-15y - 2y = -17y
This is the required middle term.

29. Factoring Trinomials Factor:
EXAMPLE
Factor:
SOLUTION
Notice that the exponent on is half that of the exponent on We will let u equal the variable to the power that is half of 6. Thus, let
This is the given polynomial, with
written as
Let Rewrite the trinomial in
terms of u.
Factor the trinomial.
Now substitute for u.
Therefore,

30. Objective #3: Example

31. Objective #3: Example

32. Objective #3: Example

33. Objective #3: Example

34. Objective #3: Example

35. Objective #3: Example

37. Factoring Using Grouping T
Factoring Trinomials
Factoring Using Grouping T
1) Multiply the leading coefficient, a, and the constant, c.
2) Find the factors of ac whose sum is b.
3) Rewrite the middle term, bx, as a sum or difference using the factors from step 2.
4) Factor by grouping.

38. Factoring Trinomials Factor by grouping: The trinomial is of the form
EXAMPLE
Factor by grouping:
SOLUTION
The trinomial is of the form
a = 1
b = 1
c = -12

39. Factoring Trinomials
CONTINUED

40. Factoring Trinomials 4) Factor by grouping. Group terms
CONTINUED
4) Factor by grouping.
Group terms
Factor from each group
Factor out a + 4, the common binomial factor
Thus,

41. Objective #4: Example

42. Objective #4: Example

43. Objective #4: Example
CONTINUED

44. Objective #4: Example
CONTINUED