1. What is Factoring?
Breaking apart a polynomial into the expressions that were MULTIPLIED to create it.
If a Polynomial can not be factored, it is called
PRIME

2. Greatest Common Factor
1. LARGEST common number
2. SMALLEST exponent on EACH letter.
3. *DIVIDE the coefficients and SUBTRACT exponents.*
Example: 15x4 + 18x3 + 30x2
Largest number: 3
Smallest exponent:2
Example: 12x4y2 + 4x3y + 20x2y5
Largest number: 4
Smallest exponent on x :2
Smallest exponent on y: 1
Example: 7xy2 + y + 14x2
GCF = PRIME
Example: 15x4 + 18x3 + 30x2
Largest number: 3
Smallest exponent:2
GCF=3x2
(3x2)(5x2+6x+10)
Now divide by 3x2:
5x2
+6x1
+10
Example: 15ax5 + 25a3x3 + 10a2x2
Largest number: 5
Smallest exponent: a1x2
GCF=5a1x2
Now divide by 5a1x2:
3x3
+5a2x
+2a1
(5ax2) (3x3 + 5a2x + 2a)
Example: 16f4g9 + 24g3 + 4fg2
Largest number: 4
Smallest exponent: g2
GCF=4g2
Now divide by 4g2:
4f4g7
+6g
+1f
(4g2) (4f4g7 + 6g + 1fg)
Largest number: 5
Example: 5h h h
Smallest exponent: 1
GCF = 5h
Now divide by 5h1:
1h3
+ 6h
- 3
(5h) (h3 + 6h – 3)

3. A value made of a number times ITSELF or x2 or x4
Difference of 2 squares
What do I look for?
Must be a BINOMIAL
Both terms are perfect squares
SUBTRACTION in the middle
A value made of a number times ITSELF or x2 or x4
a2 – b2 = (a+b)(a-b)
Square root the first term to get the first part of each answer
Square root the second term to get the 2nd part of each answer
One + and one – in the middle
16x2 - 81
( )( )
4x x
X2 - 25
( ) ( )
X X
You might need to find a GCF First!
3x2 - 75
GCF 3
(x2 – 25)
3( )( )
X X
6x3 – 24x
6x ( )
X2 - 4
6x( )( )
X X
4x3 + 8x2 – 60x
4x (x2 + 2x – 15)
(4x)(x+5)(x-3)

4. Factoring with Boxes (x + 5) (3X2 -2) (x + 7) (X2 + 2) 3X3 15x2 X3 7x2
P 46
Use this for:
4 – term polynomials
Quadratic Trinomials with a > 1
4 Term Polynomials (known as grouping)
Place the terms into the boxes in Z
Pattern.
Find the GCF of each row and column
*** Keep the sign of the closest box ***
Answer factors on top and right side
Example: x3 + 7x2 + 2x + 14
Example 2: 3x3 + 15x2- 2x - 10 x 5 X 7
3X2
3X3
15x2 X2 X3
7x2
-2x
-10 -2 2x 14 2
(x + 5)
(3X2 -2)
(x + 7)
(X2 + 2)

5. Factor by Grouping Works for 4 term polynomials!!
1. Group the first 2 terms together and the last 2 terms together.
2.Factor out a GCF for each group
3.The GCF’s combine to form one factor and the matching binomials will be the other factor.
Example: x3 + 7x2 + 2x + 14
Practice 1: 3x3 + 15x2- 2x - 10
Step 1:
(x3 + 7x2 ) +(2x + 14)
(3x3 + 15x2) (- 2x – 10)
Step 2:
X2( ) + 2( )
X+7
X+7
3x2 (x + 5) -2 (x + 5)
(3x2 -2) (x + 5)
Step 3:
(X2+ 2) ( X+7 ) x 7 x2 x3
7x2 2 2x 14

6. C C
Factors
factors
added
added b b
one
Positive
negative
one 2 3
positive 6 5
-10 3 1 6 7 2 3 5

7. C C
Factors
factors
added
added b b
one
Negative
negative
one
positive

8. Factoring Trinomials (a=1)
Standard Trinomial: aX2 + bX + c
a, b, c are numbers
Factor using 2 binomials ( )( )
First term in each binomial is x (x )(x )
Number game  Find 2 numbers that multiply to c and add to b.
Example: x2 + 5x + 4
a= b= c= 1 5 4
(x )(x ) +4 +1
Find 2 numbers that multiply to 4 and add to 5
Example: x2 + 7x + 12
(x )(x ) +3 +4
Positive C means factors have same sign
Find 2 numbers that multiply to 12 and add to 7
Example: x2 -9x + 18
(x )(x ) -6 -3
Find 2 numbers that multiply to 18 and add to -9
Example: x2 + 9x +5
(x )(x )
PRIME
Find 2 numbers that multiply to 5 and add to 9
Example: x2 + 2x - 15
(x )(x ) +5 -3
Negative C means factors have different sign
Find 2 numbers that multiply to -15 and add to 2

9. Factoring Trinomials Pg 48 Turn a TRI-nomial into 4 terms
EXAMPLE: 3x2 + 11x + 6 x 3
Fill the first and last terms into
The top left and bottom right. 3x
3x2 9x 2 2x 6
(3x+2)(x+3)
Split up the middle term into two boxes by finding the numbers that multiply to a*c
3*6 = 18 how can we split up 11x so that it multiplies to 18?
9 and 2!
Write the GCF in every row and column.
EX. 1
2x2 + 5x + 3
Ex. 2
5m2 – 17m + 6
EX. 3
6y2 – 5y - 4
EX. 4
12c2 + 11c - 5