1. Factoring Trinomials: ax2 + bx + c
Lesson 9-4
Factoring Trinomials:
ax2 + bx + c

2. Factor ax2 + bx + c
(2x + 5)( 3x + 1) = 6x2 + 2x + 15x Use the FOIL method
2  15 = 30
6  5 = 30 F O I L
Observe the following pattern in this product.
6x2 + 2x + 15x ax2 + mx + nx + c
6x2 + 17x ax2 + bx + c
= 17 and 2  15 = 6  5 m + n = b and mn = ac
Factors of 30
Sum of Factors
1,30
2,15 31 17

3. 6x2 + 17x + 5 = 6x2 + mx + nx + 5
= 6x2 + 2x + 15x + 5
= (6x2 + 2x) + (15x + 5)
= 2x(3x + 1) + 5(3x + 1)
= (3x + 1) (2x + 5)
Therefore, 6x2 + 17x + 5 = (3x + 1) (2x + 5)

4. Factor ax2 + bx + c Factor 7x2 + 22x + 3 Factors of 21 (a  c)
Sum of Factors
1,21 22

5. Factor ax2 + bx + c Factor 10x2 - 43x + 28 Factors of 280 (a  c)
Sum of Factors
-1, -280
-2, -140
-4, -70
-5, -56
-7, -40
-8, -35
-281
-142
-74
-61
-47
-43

6. Factor
5x2 + 27x + 10
24x2 - 22x + 3

7. Rule: If the GCF of the terms in a trinomial is an integer other than 1, you should first factor out this GCF.
Factor
3x2 + 24x + 45

8. Factor
4x2 + 24x + 32

9. Definition
Prime Polynomial: A polynomial that cannot be written as a product of two polynomials with integral coefficients.
Factor 2x2 + 5x -2
Factors of -4
Sum of Factors
1, -4
-1, 4
-2, 2 -3 3
There are no factors whose sum is 5. Therefore, 2x2 + 5x -2 cannot be factored using integers. Thus 2x2 + 5x -2 is a prime polynomial.

10. Factor
3x2 + 7x - 5

11. Solve by factoring
Solve. 8a2 - 9a -5 = 4 - 3a.

12. Solve by factoring
18b2 - 19b -8 = 3b2 - 5b

13. Solve Real-World Problems
At a pep rally, small foam footballs are launched by cheerleaders using a sling shot. How long is a football in the air if a student in the stands catches it on its way down 26 feet above the gym floor. Use the formula h = -16t2 + vt + s, where h = 26 feet, v = 42, and s = 6. (s is the distance above the ground when the football is shot.)
h = -16t2 + vt + s
26 = -16t2 + 42t + 6

14. Solve Real-World Problems
Mrs. Held’s science class built an air-launched model rocket for competition. When they test-launched their rocket outside the classroom, the rocket landed in a nearby tree. If the launch pad was 2 feet above the ground, the initial velocity of the rocket was 64 feet per second, and the rocket landed 30 feet above the ground, how long was the rocket in flight. Use the equation h = -16t2 + vt + s.