1. Factoring x2 + bx + c
Section 8-5

2. Goals Goal Rubric To factor trinomials of the form x2 + bx + c.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
None

4. Factoring a Trinomial x2 + bx + c
Earlier you learned how to multiply two binomials using the Distributive Property or the FOIL method. In this lesson, you will learn how to factor a trinomial into two binominals.
Notice that when you multiply (x + 2)(x + 5), the constant term in the trinomial is the product of the constants in the binomials.
(x + 2)(x + 5) = x2 + 7x + 10

5. Factoring a Trinomial x2 + bx + c
To factor a simple trinomial of the form x 2 + bx + c (leading coefficient is 1), express the trinomial as the product of two binomials. For example,
x x + 24 = (x + 4)(x + 6).
Factoring these trinomials is based on reversing the FOIL process.

6. Factoring a Trinomial x2 + bx + c
Look at the product of (x + a) and (x + b). x2 ab
(x + a)(x + b) = x2 + ax + bx + ab ax
= x2 + (a + b)x + ab bx
The coefficient of the middle term is the sum of a and b. The constant term is the product of a and b.

7. Factoring a Trinomial x2 + bx + c
Recall by using the FOIL method that
F O I L
(x + 2)(x + 4) = x 2 + 4x + 2x + 8
= x 2 + 6x + 8
To factor x 2 + bx + c into (x + one #)(x + another #), note that b is the sum of the two numbers and c is the product of the two numbers.
So we’ll be looking for 2 numbers whose product is c and whose sum is b.
Note: there are fewer choices for the product, so that’s why we start there first.

8. Procedure for Factoring Trinomials of the Form x 2 + bx + c
List the factors of 20 (two numbers that multiply to equal the constant):
Select the pairs those sum is 12 (the middle term).
Write the two
binomial factors:
Check using FOIL: 20
1×20
2×10
4×5

9. Factoring a Trinomial x2 + bx + c

10. Factoring Trinomials x2 + bx + c TIP
When c is positive, its factors have the same sign. The sign of b tells you whether the factors are positive or negative. When b is positive, the factors are positive and when b is negative, the factors are negative.

11. Example: c is Positive   (x + )(x + )
Factor each trinomial. Check your answer.
x2 + 6x + 5
(x + )(x + )
b = 6 and c = 5; look for factors of 5 whose sum is 6.
Factors of 5 Sum
1 and 
The factors needed are 1 and 5.
(x + 1)(x + 5)
Check (x + 1)(x + 5) = x2 + 5x + x + 5
Use the FOIL method.
= x2 + 6x + 5 
The product is the original trinomial.

12. Example: c is Positive (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 6x + 9
(x + )(x + )
b = 6 and c = 9; look for factors of 9 whose sum is 6.
Factors of 9 Sum
1 and 
3 and 
The factors needed are 3 and 3.
(x + 3)(x + 3)
Check (x + 3)(x + 3 ) = x2 + 3x + 3x + 9
Use the FOIL method.
= x2 + 6x + 9 
The product is the original trinomial.

13. Example: c is Positive (x + )(x + )   
Factor each trinomial. Check your answer.
x2 – 8x + 15
(x + )(x + )
b = –8 and c = 15; look for factors of 15 whose sum is –8.
Factors of –15 Sum
–1 and –15 –16 
–3 and –5 –8 
The factors needed are –3 and –5 .
(x – 3)(x – 5)
Check (x – 3)(x – 5 ) = x2 – 5x – 3x + 15
Use the FOIL method.
The product is the original trinomial.
= x2 – 8x + 15 

14. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 8x + 12
(x + )(x + )
b = 8 and c = 12; look for factors of 12 whose sum is 8.
Factors of Sum
1 and 
2 and 
The factors needed are 2 and 6 .
(x + 2)(x + 6)
Check (x + 2)(x + 6 ) = x2 + 6x + 2x + 12
Use the FOIL method.
= x2 + 8x + 12 
The product is the original trinomial.

15. Your Turn: (x + )(x+ )    Factor each trinomial. Check your answer.
b = –5 and c = 6; look for factors of 6 whose sum is –5.
Factors of 6 Sum
–1 and –6 –7 
–2 and –3 –5 
The factors needed are –2 and –3.
(x – 2)(x – 3)
Check (x – 2)(x – 3) = x2 – 3x – 2x + 6
Use the FOIL method.
= x2 – 5x + 6 
The product is the original trinomial.

16. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 13x + 42
(x + )(x + )
b = 13 and c = 42; look for factors of 42 whose sum is 13.
Factors of 42 Sum
1 and 
6 and 7 13 
2 and
The factors needed are 6 and 7.
(x + 6)(x + 7)
Check (x + 6)(x + 7) = x2 + 7x + 6x + 42
Use the FOIL method.
= x2 + 13x + 42 
The product is the original trinomial.

17. Your Turn: (x + )(x+ )    Factor each trinomial. Check your answer.
b = –13 and c = 40; look for factors of 40 whose sum is –13.
(x + )(x+ )
Factors of 40 Sum 
–2 and –20 –22
–4 and –10 –14 
–5 and – –13
The factors needed are –5 and –8.
(x – 5)(x – 8)
Check (x – 5)(x – 8) = x2 – 8x – 5x + 40
Use the FOIL method.
= x2 – 13x + 40 
The product is the original polynomial.

18. Factoring Trinomials x2 + bx + c TIP
When c is negative, its factors have opposite signs. The sign of b tells you which factor is positive and which is negative. The factor with the greater absolute value has the same sign as b.

19. Example: c is Negative (x + )(x + )   Factor each trinomial.
b = 1 and c = –20; look for factors of –20 whose sum is 1. The factor with the greater absolute value is positive.
Factors of –20 Sum 
–1 and
–2 and 
–4 and
The factors needed are 5 and –4.
(x – 4)(x + 5)

20. Example: c is Negative (x + )(x + )   Factor each trinomial.
b = –3 and c = –18; look for factors of –18 whose sum is –3. The factor with the greater absolute value is negative.
Factors of –18 Sum 
1 and – –17
2 and – – 7 
3 and – – 3
The factors needed are 3 and –6.
(x – 6)(x + 3)

21. If you have trouble remembering the rules for which factor is positive and which is negative, you can try all the factor pairs and check their sums.
Helpful Hint

22. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 + 2x – 15
b = 2 and c = –15; look for factors of –15 whose sum is 2. The factor with the greater absolute value is positive.
(x + )(x + )
Factors of –15 Sum 
–1 and
–3 and 
The factors needed are –3 and 5.
(x – 3)(x + 5)
Check (x – 3)(x + 5) = x2 + 5x – 3x – 15
Use the FOIL method.
= x2 + 2x – 15 
The product is the original polynomial.

23. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 – 6x + 8
(x + )(x + )
b = –6 and c = 8; look for factors of 8 whose sum is –6.
Factors of 8 Sum 
–1 and – –7
–2 and – –6 
The factors needed are –4 and –2.
(x – 2)(x – 4)
Check (x – 2)(x – 4) = x2 – 4x – 2x + 8
Use the FOIL method.
= x2 – 6x + 8 
The product is the original polynomial.

24. Your Turn: (x + )(x + )   
Factor each trinomial. Check your answer.
x2 – 8x – 20
(x + )(x + )
b = –8 and c = –20; look for factors of –20 whose sum is –8. The factor with the greater absolute value is negative.
Factors of –20 Sum 
1 and – –19
2 and – –8 
(x – 10)(x + 2)
The factors needed are –10 and 2.
Check (x – 10)(x + 2) = x2 + 2x – 10x – 20
Use the FOIL method.
= x2 – 8x – 20 
The product is the original polynomial.

25. Practice

26. More Practice

27. Assignment
8-5 Exercises Pg : #10 – 46 even