1. Factoring trinomials: x2 + bx + c
Lesson 72
Factoring trinomials: x2 + bx + c

2. Factoring trinomials The polynomial x2 + bx + c is a trinomial
Like numbers, some trinomials can be factored into 2 linear factors.

3. Factoring when c is positive
Factor x2 + 9x + 18
In this trinomial b is 9 and c is 18
Because b is positive, it must be the sum of 2 positive numbers that are factors of c
List all factors of c, which is 18:
1,18 2,9 3,6
Which pair adds up to b , which is 9?
3,6
So x2 +9x + 18 = (x+3)(x+6)

4. practice Factor x2 + 5x + 6 x2 + 8x + 15 x2 + 4x + 4 x2 + 4x + 3

5. Factoring when c is positive and b is negative
Factor x2-5x+4
List factors of 4: 1,4 2,2 -1, ,-2
Which pair adds up to -5 ?
-1,-4
x2 -5x + 4 = (x-1)(x-4)

6. practice Factor: x2 – 8x + 15 x2 -7x + 12 x2 -5x + 6 x2 – 4x + 3

7. Factoring when c is negative
Factor x2 + 3x -10
List factors of -10
-1, ,1 -2,5 -5,2
Which pair adds up to +3?
-2,5
x2 + 3x -10 = (x-2)(x+5)

8. practice
Factor: x2 – 7x – 8
x2 – 7x – 30
x2 + 3x -54
x2 + 5x - 14

9. Factoring with 2 variables
x2 + 5xy + 6y2
b = 5y c = 6y2
list factors of 6y2
1,6y2 6,y2 2,3y2 3,2y2 y,6y 2y,3y
Which pair adds up to 5y
2y, 3y
So x2 + 5xy + 6y2 = (x+2y)(x+3y)

10. factor
x2 + 2xy – 3y2
x2 + 4xy +4y2
x2 -7xy -18y2

11. Rearranging terms before factoring
-21-4x+x2
Write the trinomial in standard form , then factor
x2 -4x-21
list factors of -21, that add up to -4
-1,21 1, ,7 3,-7
3, -7 adds up to -4
so x2-4x-21 = (x+3)(x-7)

12. practice
Factor: x + x2
3x – 10 + x2
-8 + x2 – 7x

13. Evaluating trinomials
Evaluate x2+5x-14 and its factors
for x = 3
x2 + 5x -14 = (x+7)(x-2)
(3)2 +5(3)-14 = (3+7)(3-2)
= (10)(1)
= 10

14. practice
Evaluate
x2 + 5x – for x = 2

15. Lesson 75
The pattern used for trinomials is different when a is not 1.
Factor 2x2 + 7x + 5
2x2 has factors 2x and x so we know
(2x )( x )
Factors of 5 are 1, ,-5
Because the middle term is positive, we can eliminate the -1,-5 pair
Check each other pair to see if the middle terms added equal the middle term in the trinomial
(2x + 1)(x+5) or (2x +5)(x+1)
2x2 +10x+x x2+2x+5x+5
2x2 +11x x2 + 7x + 5

16. factor
3x2 +13x+12
6x2 +11x+3

17. Factoring when b is negative and c is positive
Factor 6x2 - 11x + 3
factors of 6x2 (x )(6x ) or (2x )(3x )
Factors of ,3 or -1,-3
Since the middle term is negative, we can eliminate 1,3
Try (x-1)(6x-3) = 6x2-3x-6x+3=6x2-9x+3
(x-3)(6x-1)=6x2-x-18x+3=6x2-19x+3
(2x-1)(3x-3)=6x2-6x-3x+3=6x2-9x+3
(2x-3)(3x-1)=6x2-2x-9x+3=6x2-11x+3

18. practice
4x2 -23x + 15
10x2-23x+12

19. Factoring when c is negative
4x2 + 4x-3
Factors of 4x (4x )(x )or(2x )(2x )
Factors of , -3 or -1,3
(4x+1)(x-3)= 4x2-11x-3
(4x-3)(x+1)= 4x2+x-3
(4x-1)(x+3)= 4x2+11x-3
(4x+3)(x-1)= 4x2-x-3
(2x+1)(2x-3)= 4x2-4x-3
(2x-1)(2x+3)=4x2+4x-3

20. factor
6x2+8x-8
2x2-3x-20
3x2-11x-4

21. Factoring with 2 variables
Factor 2x2-11xy+5y2
Factors of 2x2 (2x )(x )
Factors of 5y2 5y, y or -5y, -y
Since the middle sign is negative , we can eliminate the 5y,y
(2x-5y)(x-y) or (2x-y)(x-5y)
2x2-2xy-5xy+5y2 or 2x2-10xy-xy+5y2
2x2-7xy+5y or 2x2-11xy+5y2

22. practice
Factor 2x2-5xy+2y2
6x2+11xy+4y2

23. Rearranging before factoring
Trinomials must be in standard form before factoring

24. Factoring trinomials by using the GCF
Lesson 79
Factoring trinomials by using the GCF

25. Factoring trinomials with positive leading coefficients
Always factor the GCF first, then continue factoring as in previous lessons
x4+5x3+6x2
GCF is x2, so
x2(x2+5x+6)
x2(x+3)(x+2)

26. Practice
Factor
4x3-4x2-80x
a5+3a4-18a3
2m4-16m3+30m2

27. Factoring with negative lead coefficients
It will always be easier to factor a trinomial with a positive lead coefficient, so whenever possible, factor out a negative if your trinomial has a negative lead coefficient.
-x2+x+56= -1(x2-x-56)=-1(x-8)(x+7)