1. Factoring ax2 + bx + c
Section 8-6

2. Goals Goal Rubric To factor trinomials of the form ax2 + 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 ax2 + bx + c
In the previous lesson you factored trinomials of the form x2 + bx + c. Now you will factor trinomials of the form ax + bx + c, where a ≠ 0.

5. THE FUN WAY TO FACTOR TRINOMIALS
X-BOX Method
THE FUN WAY TO FACTOR TRINOMIALS

6. X-Box Method
This is a guaranteed method for factoring trinomials of the form ax2 + bx +c — no guessing necessary!
We will learn how to factor quadratic equations using the x-box method
Background knowledge needed:
Basic x-factor problems
General form of a trinomial
The factoring box

7. m n mn m n m+n The X-Factor Given m & n on the sides of the x factor
m•n is the top of the x factor m n
m+n
m+n is the bottom of the x factor

8. The X-Factor mn n m
m+n

9. What’s on Top & Bottom? 5 3
Given m & n on the sides of the x factor ?
m•n is the top of the x factor 5 3 ?
m+n is the bottom of the x factor

10. 5 3 15 5 3 8 The X-Factor Given m & n on the sides of the x factor
m•n is the top of the x factor 5 3 8
m+n is the bottom of the x factor

11. The X-Factor -3 9 ? -3 9 ?

12. The X-Factor -3 9
-27 -3 9 6

13. The X-Factor 12
Given m•n 7
Given m+n 12 ? ?
Find m & n 7

14. The X-Factor 12
Given m•n 7
Given m+n 12 3 4
Find m & n 7

15. The X-Factor 8
Given m•n 9
Given m+n 8 ? ?
Find m & n 9

16. The X-Factor 8
Given m•n 9
Given m+n 8 8 1
Find m & n 9

17. The X-Factor
-84
Given m•n -5
Given m+n
-84
Find m & n -5

18. To solve a difficult X-Factor START AT 1 AND CHECK THE SUM
-84
-84
Given m•n
= -83
-41 = -39
-28 = -25
-21 = -17
-14 = -8
= -5 -5
Given m+n
-84
Find m & n
-12 7 -5

19. Why Know the “X” Factor?

20. Why Know the “X” Factor? The “X” factor is a Simple way to
FACTOR TRINOMIALS

21. General Form of a Trinomial
c is COEFFICIENT
of last term
b is COEFFICIENT
of x
a is COEFFICIENT
of x2

22. Identify a, b & c
b is 3
c is 9
a is 5

23. Factor the X-Box Method
ax2 + bx + c
Combine the x-factor and a 2⨯2 box to create the x-box method.
Factor Col. 1
Factor Col. 2
Product
ac=mn
First and Last Coefficients
1st Term
GCF Row 1
Factor n n m
Middle
Last term
Factor m
Factor Row 2
b=m+n
Sum

24. X-Box Procedure ax2 + bx + c 1st Term Factor n Factor m Last term
Create an x-factor with the product ac on the top, the middle term b on the bottom and the factors m & n on the sides.
Create a 2x2 box.
In the top left, put the 1st term. In the bottom right corner, put the last term.
Put the two factors m and n times the variable x in the open boxes.
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial.
1st Term
Factor n
Factor m
Last term

25. Factoring ax2+bx+c
Factor 5x2+11x+2
5•2= 10
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 1 10 11 x 2 5x x
10x
5x2 1 2
(x+2)(5x+1)

26. Factor 3x2 – x – 4 3 -4 -1 3x -4 x 3x -4x 3x2 1 -4 3•-4= -12
Factoring ax2+bx+c
Factor 3x2 – x – 4
3•-4=
-12
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 3 -4 -1 3x -4 x 3x
-4x
3x2 1 -4
(x+1)(3x – 4)

27. Factor 12x2 +5x – 2 8 -3 5 4x -1 3x 12x2 8x -3x 2 -2 12•-2= -24
Factoring ax2+bx+c
Factor 12x2 +5x – 2
12•-2=
-24
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 8 -3 5 4x -1 3x
12x2 8x
-3x 2 -2
(3x+2)(4x – 1)

28. Factor 15x2 +7x – 2 10 -3 7 5x -1 3x 10x -3x 15x2 2 -2 15•-2= -30
Factoring ax2+bx+c
Factor 15x2 +7x – 2
15•-2=
-30
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 10 -3 7 5x -1 3x
10x
-3x
15x2 2 -2
(3x+2)(5x – 1)

29. Factoring ax2+bx+c
Factor 10x2 +9x +2
10•2= 20
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 5 4 9 5x 2 2x 5x 4x
10x2 1 2
(2x+1)(5x +2)

30. Your Turn:
Factor: 2x² - 13x + 6

31. Factor 2x2 – 13x +6 -12 -1 -13 2x -1 x -12x -x 2x2 -6 6 2•6= 12
Factoring ax2+bx+c
Factor 2x2 – 13x +6
2•6= 12
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial.
-12 -1
-13 2x -1 x
-12x -x
2x2 -6 6
(2x – 1)(x – 6)

32. Your Turn:
Factor: 4x² + 11x – 3

33. Factor 4x²+11x – 3 12 -1 11 4x -1 x 12x -x 4x2 3 -3 4•-3= -12
Factoring ax2+bx+c
Factor 4x²+11x – 3
4•-3=
-12
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 12 -1 11 4x -1 x
12x -x
4x2 3 -3
(4x – 1)(x + 3)

34. Your Turn:
Factor: 3x²+8x+4

35. Factor 3x²+8x+4 6 2 8 3x 2 x 6x 2x 3x2 2 4 4•3= 12 Factoring ax2+bx+c
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 6 2 8 3x 2 x 6x 2x
3x2 2 4
(3x + 2)(x + 2)

36. Your Turn:
Factor: -5x²+6x-1

37. Factor -5x²+6x-1 5 1 6 -5x 1 x 5x x -5x2 -1 -1 -5•-1= 5
Factoring ax2+bx+c
Factor -5x²+6x-1
-5•-1= 5
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. 5 1 6
-5x 1 x 5x x
-5x2 -1 -1
(-5x + 1)(x – 1)

38. Factor Completely * Always Factor out the GCF First *
To factor a polynomial completely, first factor at the GCF of the polynomial’s terms.
Then factor the remaining polynomial until it is written as the product of polynomials that cannot be factored further.
* Always Factor out the GCF First *

39. Example: Factor Completely
Factor Completely: 18x2 – 33x + 12 First factor out the GCF 3 3(6x2 – 11x + 4) Then factor the remaining polynomial (6x2 – 11x + 4)

40. Factor 6x2 – 11x+4 -8 -3 -11 2x -1 3x -8x -3x 6x2 -4 4 6•4= 24
Factoring ax2+bx+c
Factor 6x2 – 11x+4
6•4= 24
Set up & Solve X-factor:
Put ac on top
Put b on bottom
Determine side factors
Set up & Solve “BOX”:
Put First Term in First Box
Put Last Term in Last Box
Put Side Factors Times x in Box
Determine the GCF of the upper boxes and the remaining top & side factors.
The sum of the factor’s for the top and the side are the factors of the polynomial. -8 -3
-11 2x -1 3x
-8x
-3x
6x2 -4 4
(3x – 4)(2x – 1)

41. Example: Factor Completely
Factor Completely: 18x2 – 33x + 12
First factor out the GCF 3
3(6x2 – 11x + 4)
Then factor the remaining polynomial (6x2 – 11x + 4)
(3x – 4)(2x – 1)
Completely Factored: 3(3x – 4)(2x – 1)

42. Your Turn: Completely factor 8x2 – 36x – 20 4(2x + 1)(x – 5)

43. Assignment
8-6 Exercises Pg : #8 – 58 even