1. How to Factor!

2. 1) Greatest common factor
Always work in the following order…
1) Greatest common factor
2) Perfect Trinomial Square
3) Quadratic functions
4) Difference of two Squares…
5) By grouping

3. 1) Greatest common factor
Look for…
A constant,
A variable,
Or a combination of both.
Ex 1: Factor 7x+7 ……
Ex 2: Factor 3x+x² …..
Ex 3: Factor 2x² - 8x…

4. 2) Perfect Trinomial Square x² + 8x + 16 or x² - 8x +16
Is the first term a perfect square?
Is the last term a perfect square?
Is the middle term…
2· =2 ·x ·4 ?
If so, it factors as…
Ex 1: x² +8x+ 16 = (x + 4)2 Use the middle sign
Ex 2: x² -8x = (x - 4)2 Use the middle sign

5. 2) Quadratic Functions Ex: x 2 + 10x + 16 Possible Products
a) Middle term is positive
Ex: x x Possible Products
16 = 1 x 16
=2 x 8
= 4 x 4
But Middle term must add up to +10!
2 +8=10
Factors as : (x + 2)(x + 8)

6. b) Middle term is negative
Ex: x x Possible Products
16 = 1 x 16
= 2 x 8
= 4 x 4
But Middle term must add up to -10!
= -10
Factors as : (x - 2)(x - 8)

7. That’s fine… BUT what if a≠1 Like …2x 2 - 5x – 12 ?

8. Bustin’ da middle term!! …
Ex: 2x 2 - 5x – 12
Multiply (2)(-12) = -24
Possible Factors:
= -1 x 24
= -2 x 12
= 3 x 8
= -4 x 6
Yet… the Middle term is (- 5x)
so, we need more negatives!
3x + - 8x and…

9. Bustin’ da Middle!! Cont…
2x 2 - 5x Original
2x 2 + 3x + -8x Bust Middle.
(2x 2 + 3x) + (-8x-12) Group.
x(2x+3) – 4(2x+3) Factor GCF.
(2x+3) (x-4)
(Write what they have in common), then (write what’s left).

10. On your own… Yet…the middle term is(- 8x) ,so we need more negatives!
Ex: 6x 2 - 8x – 8
Multiply (6)(-8) = -48
Possible Factors:
= -1 x 48
= -2 x 24
= -3 x16
= 4 x 12
= -6 x 8
Yet…the middle term is(- 8x) ,so we need more negatives!
-12x + 4x so…

11. Bustin’ da Middle!! Cont…
6x 2 - 8x Original
6x x + 4x -8 Bust Middle.
(6x 2 -12x) + (4x-8) Group.
6x(x-2) + 4(x-2) Factor GCF.
(x-2) (6x+4)
(Write what they have in common), then (write
what’s left).

12. 3a) Difference of two Squares x² - 16
Is the first term a perfect square?
Is the second term a perfect square?
Is there a minus between the two terms?
Are there only two terms?...if yes
Factor as:
( )( – )
Ex: x = (x+4)(x-4)

13. 3b) Difference of 2 cubes a 3- b 3 = (a - b)(a 2+ ab + b 2)
Factor: x
Factors as…

14. 3c) Sum of 2 cubes a 3+ b 3 = (a + b)(a 2 - ab + b 2)
Factor: x
Factors as…

15. 3d) Difference of 4th powers is the “hidden” form of the difference of two squares
a 4- b 4 = =(a 2 - b 2)(a 2+ b 2)
Factor…
Factors as…
= (a - b)(a + b)(a 2+ b 2)

16. 5) Grouping Used when there are 4 or more terms. Ex: 2ab + 14a + b + 7
(2ab +14a) + (b + 7) Group
2a(b+7) + 1(b+7) … factor, also 1
(b+7) (2a+1)
(Write what they have in common), then
(write what’s left).

17. Grouping cont…. Ex: xy + 3x – y²- 3y
(xy + 3x) + (-y²-3y)… Always use a plus sign
between parentheses,
x(y+3) - y(y+3) then factor GCF
(y+3) (x-y)
(Write what they have in common), then (write
what’s left).

18. Finding the zero’s (or solving)
Try to factor…
His implies that either expression must be 0
There are TWO solutions: x=6; x=-5

19. Something different…
Solve by substitution.

20. Using Synthetic Division
Factoring
Using Synthetic Division

21. Synthetic Division
We can use this method as long as you know one of the zero’s,
Place the zero’s value of x here. 2
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Bring first number down below line 2
Add these up
- 4 2
Add these up
Add these up 1 -2 1
This is the remainder
List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.
Put variables back in (one x was divided out in the process, so the first term is x²).
Is the factored form

22. You Try to factor! - 2 4 8 -25 -50 - 8 50 4 x2 + x - 25
set the factor = 0 and solve for x, this will be your divisor.
- 2
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Bring first number down below line
- 8
Add these up 50
Add these up
Add these up
No remainder so x + 2 IS a factor because it divided in evenly 4
x2 + x
- 25
So the answer is the divisor times the quotient:
List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.
Put variables back in (one x was divided out in the process, so the first term is x²).
You could factor even further…

23. You can use synthetic division to divide polynomials
Set divisor = 0 and solve. Put answer here.
x + 3 = 0 so x = - 3 1
- 3
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Bring first number down below line
- 3
Add these up
- 9 3
Add these up
Add these up 1
x2 + x 3
- 1 1
This is the remainder
List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0.
Is the quotient
Put variables back in (one x was divided out in the process, so the first term is x²).

24. CAVEAT…… Find the quotient
0 x3
0 x
Set divisor = 0 and solve. Put answer here.
x - 4 = 0 so x = 4 1 4
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Multiply these and put answer above line in next column
Bring first number down below line 4
Add these up 16
Add these up 48
Add these up
192
Add these up 1
x3 + x x + 4 12 48
198
This is the remainder
List all coefficients (numbers in front of x's) and the constant along the top. Don't forget the 0's for missing terms.
Now put variables back in (remember one x was divided out in the process so first term is x3).
Is the quotient