1. Factoring

2. Factoring by Greatest Common
Factor (GCF)

3. Factoring by removing the GCF “undoes” the
multiplication. (It’s the distributive
property in reverse.)
2x2 – 4
2x2 – (2  2)
2(x2 – 2)
Check 2(x2 – 2)
2x2 – 4

4. Example
8x3 – 4x2 – 16x
2x2(4x) – x(4x) – 4(4x)
4x(2x2 – x – 4)
4x(2x2 – x – 4)
8x3 – 4x2 – 16x
Check


5. Factoring ax2+bx+c

6. 2
x bx c 
To factor, find two numbers that:
Add to b Multiply
To c

7. Example
Factor x2 + 15x + 36. Check your answer.
(x + )(x + )
(x + 1)(x + 36) = x2 + 37x + 36
(x + 2)(x + 18) = x2 + 20x + 36
(x + 3)(x + 12) = x2 + 15x + 36




8. Example
Factor each trinomial. Check your answer.
x2 + 10x + 24
(x + )(x + )
(x + 4)(x + 6) = x2 + 10x + 24

9. Factor each trinomial.
x2 + x – 20
(x + )(x + )
(x – 4)(x + 5)

10. Remember to look for a GCF first!
3x2 + 18x – 21
3(x + )(x + )
3(x – 1)(x + 7)
3(x2 + 6x – 7)

11. Factor each trinomial.
x2 – 11xy + 30y2
(x – 5y)(x – 6y)

12. A trinomial is a perfect square if:
• The first and last terms are perfect squares.
• The middle term is double the product of
the square roots.
9x2 + 12x + 4
3x 3x 2(3x 2) 2 2• ••
REMEMBER
THE X!
Perfect Square Trinomials

13. Factor.
81x2 + 90x + 25
Example
(9x)(9x) (5)(5)
The middle term = 2(9x)(5), so this is
a perfect square trinomial

14. Factor.
Example
2
81 90 25
(9 )(9 )
x x
x x
 
 

15. Factor.
Example
2
81 90 25
(9 5 )(9 5)
x x
x x
 
 

16. A polynomial is a difference of two squares if:
•There are two terms, one subtracted from the
other.
• Both terms are perfect squares.
4x2 – 9
2x 2x 3 3
For variables:
All even powers are perfect squares
Difference of Two Squares

17. Example
2
49 100x 
(7x)( 7x) (10)( 10)

18. Example
2
49 100
(7 10)(7 10)
x
x x

 
Center term cancels out, so use opposite signs.

19. Factoring ax2 + bx + c

20. The “box” method
2
2x
6
First
term
Last
term
Include the
signs!!!
2
2 7 6x x 

21. Multiply a and c. (In this case, that would be 2 x 6 = 12) Put the
factors of “ac” that add up to “b” in the other squares, with their signs
and an “x”. Order does not matter.
2
2x
6
One
factor
The
other
factor
3x
4x
2
2 7 6x x 
4 and 3 are factors
of 12 that add up to
7, so they go in the
empty spaces. Add
an x because they
represent the
middle term.

22. Factor the GCF out of each row or
column. Use the signs from the closest
term.
2
2x
63x
4xGCF is +2x

23. Factor the GCF out of each row or
column. Use the signs from the closest
term.
2
2x
63x
4x+2x
GCF is +3

24. Factor the GCF out of each row or
column. Use the signs from the closest
term.
2
2x
63x
4x
GCF is +x
+2x
+3

25. Factor the GCF out of each row or
column. Use the signs from the closest
term.
2
2x
63x
4x
+x GCF =+2
+2x
+3

26. The “outside” factors combine to factor
the quadratic.
2
2x
63x
4x
+x +2
+2x
+3
(2x+3)(x+2)

27. You can also use trial and error. Determine the
possible factors for the first and last terms, and
then keep trying combinations until you find the
one that works.
2
3 17 10 x x
First term: 3x and x are the only possible
factors
Last term: Factors are 1 and 10 or 2 and 5

28. 2
(3 1)( 10) 3 31 10    x x x x
2
(3 10)( 1) 3 13 10    x x x x
2
(3 5)( 2) 3 11 10    x x x x
2
(3 2)( 5) 3 17 10    x x x x
Only this combination
works.

29. Factoring by Grouping

30. Factoring by grouping:
If you have four terms –
make 2 groups of 2 and factor out
the GCF from each.
MUST be used on 4 terms
CAN be used on 3.

31. Example
Factor each polynomial by grouping.
Check your answer.
6h4 – 4h3 + 12h – 8
(6h4 – 4h3) + (12h – 8)
2h3(3h – 2) + 4(3h – 2)
2h3(3h – 2) + 4(3h – 2)
(3h – 2)(2h3 + 4)

32. Factor each polynomial by grouping.
Check your answer.
Check (3h – 2)(2h3 + 4)
3h(2h3) + 3h(4) – 2(2h3) – 2(4)
6h4 + 12h – 4h3 – 8
6h4 – 4h3 + 12h – 8

33. Example
Factor each polynomial by grouping.
Check your answer.
5y4 – 15y3 + y2 – 3y
(5y4 – 15y3) + (y2 – 3y)
5y3(y – 3) + y(y – 3)
5y3(y – 3) + y(y – 3)
(y – 3)(5y3 + y)

34. Factor each polynomial by grouping.
Check your answer.
5y4 – 15y3 + y2 – 3y
Check (y – 3)(5y3 + y)
y(5y3) + y(y) – 3(5y3) – 3(y)
5y4 + y2 – 15y3 – 3y
5y4 – 15y3 + y2 – 3y 

35. You can use factoring by grouping on trinomials.
2
3 11 10x x 
Split the 11x into two terms
(coefficients should multiply to 30,
because 3x10=30)

36. 2
3 6 5 10x x x  
Write the terms in whichever
order will allow you to group.

37. 2
(3 6 ) (5 10)
3 ( 2) 5( 2)
(3 5)( 2)
x x x
x x x
x x
  
  
 