1. Factoring Quadratic Expressions

2. Specific Expressions Trinomial – Binomial – Monomial –
Consisting of three terms (Ex: 5x3 – 9x2 + 3)
Consisting of 2 terms (Ex: 2x6 + 2x)
Consisting of one term (Ex: x2)

3. ax2 + bx +c Quadratic Expression
An expression in x that can be written in the standard form:
ax2 + bx +c
Where a, b, and c are any number except a ≠ 0.

4. Factoring
The process of rewriting a mathematical expression involving a sum to a product. It is the opposite of distributing.
Example:
SUM
PRODUCT

5. (Remember that 3 and 4 are factors of 12 since 3.4=12)
If x2 + 8x + 15 = ( x + 3 )( x + 5 )
then
x + 3 and x + 5 are
called factors of x2 + 8x + 15
(Remember that 3 and 4 are factors of 12 since 3.4=12)

6. Finding the Dimensions of a Generic Rectangle
Mr. Wells’ Way to find the product for a generic rectangle:
Make sure to Check
Second, find missing WHOLE NUMBER dimensions on the individual boxes.
First, find the POSITIVE Greatest Common Factor of two terms in the bottom row. 5
10x
-15 2x
4x2
-6x 2x -3
Lastly, write the answer as a Product:

7. A Pattern with Generic Rectangles
The product of one diagonal always equals the product of the other diagonal.
Example:
10x . -6x = -60x2
10x
-15
4x = -60x2
4x2
-6x

8. Factoring with the Box and Diamond
c is always in the top right corner
Because of our pattern, the missing boxes need to multiply to:
Fill in the results from the diamond and find the dimensions of the box: 3 3x 4x +6
(2x2)(6) 12x2
GCF 2x
2x2
4x x
___
Diamond Problem 7x x 2
ax2 is always in the bottom left corner
Write the expression as a product:
The missing boxes also have to add up to bx in the sum
( 2x + 3 )( x + 2 )

9. Factoring Example Factor: (3x2)(-10) -30x2 5 15x -2x -10 -2x 15x GCF x
Product c
(3x2)(-10)
-30x2 5
15x
-2x
-10
ax2c
-2x x
GCF x
3x2
___ bx
ax2
13x 3x -2
Sum
( x + 5 )( 3x – 2 )

10. Factoring: Different Order
Rewrite in Standard Form: ax2 + bx + c
Product c
(15x2)(-77)
-1155x2 11
33x
-35x
-77
ax2c
-35x x
GCF 5x
15x2
___ bx
ax2
-2x 3x -7
Sum
( 5x + 11 )( 3x – 7 )

11. Factoring: Perfect Square
Product c
(x2)(9)
9x2 3 3x 9
ax2c
3x x
GCF x x2
___ bx
ax2 6x x 3
Sum
( x + 3 )( x + 3 )
( x + 3 )2

12. Factoring: Missing Terms
Product c
(9x2)(-4)
-36x2 2 6x
-6x -4
ax2c
-6x x
GCF 3x
9x2
___ bx
ax2 3x -2
Sum
( 3x + 2 )( 3x – 2 )

13. Factoring: Which Expression is correct?
Notice that every term is divisible by 2
Factor:
If you use the box and diamond, the following products are possible: x2 ÷2
Which is the best possible answer?

14. Factoring: Factoring Completely
Ignore the GCF and factor the quadratic
Reverse Box to factor out the GCF
5( x + 3 )( 2x – 1 )
Product
Don’t forget the GCF c
(2x2)(-3)
-6x2 3 6x -x -3
ax2c
-x x
GCF x
2x2
___ bx
ax2 5x 2x -1
Sum

15. Factoring: Factoring Completely
Ignore the GCF and factor the quadratic
Reverse Box to factor out the GCF
3x(x + 3)(x – 5)
Product
Don’t forget the GCF c
(x2)(-15)
-15x2 3 3x
-5x
-15
ax2c
-5x x
GCF x x2
___ bx
ax2
-2x x -5
Sum

16. Factoring: Forgot to Factor a GCF
No, it is not factored completely because one of the factors still has a GCF bigger than 1.
When factoring the above expression a student came up with the following answer. Is it factored completely?
Factor out the GCF with a reverse box
Substitute the result:
NOTE: I do not recommend relying on this. It can be used IF you forget to check for a GCF.

17. Factoring: Just Factoring a GCF
Reverse Box to factor out the GCF
There is no longer a quadratic, it is not possible to factor anymore. There is not always more factoring after the GCF.

18. Factoring: Ensuring “a” is Positive
When the x2 term is negative, it is difficult to factor.
Reverse Box to factor out the negative
Ignore the GCF and factor the quadratic
-( x + 6 )( x + 7 )
Product
Don’t forget the GCF c
(x2)(42)
42x2 6 6x 7x 42
ax2c
6x x
GCF x x2
___ bx
ax2
13x x 7
Sum