The document shows examples of advanced factoring techniques including factoring by grouping, factoring quadratics, and factoring a polynomial into linear factors in order to graph the solution set. It provides step-by-step workings to factor expressions into their simplest linear factors in order to determine possible solutions for equations.

1. Section 9-9
Advanced Factoring Techniques

2. Warm-up
Factor.
1. 2x + 3x + x
3 2
2. 2x + 2x + x + x
3 2 2

3. Warm-up
Factor.
1. 2x + 3x + x
3 2
2. 2x + 2x + x + x
3 2 2
x(2x + 3x + 1)
2

4. Warm-up
Factor.
1. 2x + 3x + x
3 2
2. 2x + 2x + x + x
3 2 2
x(2x + 3x + 1)
2
x(2x + 1)( x + 1)

5. Warm-up
Factor.
1. 2x + 3x + x
3 2
2. 2x + 2x + x + x
3 2 2
(2x + 2x ) + ( x + x )
3 2 2
x(2x + 3x + 1)
2
x(2x + 1)( x + 1)

6. Warm-up
Factor.
1. 2x + 3x + x
3 2
2. 2x + 2x + x + x
3 2 2
(2x + 2x ) + ( x + x )
3 2 2
x(2x + 3x + 1)
2
2x ( x + 1) + x( x + 1)
2
x(2x + 1)( x + 1)

7. Warm-up
Factor.
1. 2x + 3x + x
3 2
2. 2x + 2x + x + x
3 2 2
(2x + 2x ) + ( x + x )
3 2 2
x(2x + 3x + 1)
2
2x ( x + 1) + x( x + 1)
2
(2x + x )( x + 1)
2
x(2x + 1)( x + 1)

8. Warm-up
Factor.
1. 2x + 3x + x
3 2
2. 2x + 2x + x + x
3 2 2
(2x + 2x ) + ( x + x )
3 2 2
x(2x + 3x + 1)
2
2x ( x + 1) + x( x + 1)
2
(2x + x )( x + 1)
2
x(2x + 1)( x + 1)
x(2x + 1)( x + 1)

9. Factoring by Grouping

10. Factoring by Grouping
Grouping terms with common factors, then factoring out the
GCF and applying distribution

11. Factoring by Grouping
Grouping terms with common factors, then factoring out the
GCF and applying distribution
...you know, like we do with factoring quadratics

12. Example 1
Factor
6a b + 2ac + 3ab + bc
2 2

13. Example 1
Factor
6a b + 2ac + 3ab + bc
2 2
(6a b + 2ac) + (3ab + bc)
2 2

14. Example 1
Factor
6a b + 2ac + 3ab + bc
2 2
(6a b + 2ac) + (3ab + bc)
2 2
2a(3ab + c) + b(3ab + c)

15. Example 1
Factor
6a b + 2ac + 3ab + bc
2 2
(6a b + 2ac) + (3ab + bc)
2 2
2a(3ab + c) + b(3ab + c)
(3ab + c)(2a + b)

16. Example 2
Factor
4x + 4x − 15
2

17. Example 2
Factor
4x + 4x − 15
2
4x + 10x − 6x − 15
2

18. Example 2
Factor
4x + 4x − 15
2
4x + 10x − 6x − 15
2
(4x + 10x ) + (−6x − 15)
2

19. Example 2
Factor
4x + 4x − 15
2
4x + 10x − 6x − 15
2
(4x + 10x ) + (−6x − 15)
2
2x(2x + 5) − 3(2x − 5)

20. Example 2
Factor
4x + 4x − 15
2
4x + 10x − 6x − 15
2
(4x + 10x ) + (−6x − 15)
2
2x(2x + 5) − 3(2x − 5)
(2x − 3)(2x + 5)

21. Factoring by Grouping and
Graphing

22. Factoring by Grouping and
Graphing
Factoring into linear factors and graphing to see possible
solutions

23. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2

24. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2

25. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2
x ( x + 3) − y( x + 3) = 0
2

26. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2
x ( x + 3) − y( x + 3) = 0
2
( x − y)( x + 3) = 0
2

27. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2
x ( x + 3) − y( x + 3) = 0
2
( x − y)( x + 3) = 0
2
x − y=0
2

28. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2
x ( x + 3) − y( x + 3) = 0
2
( x − y)( x + 3) = 0
2
x − y=0
2
x+3=0

29. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2
x ( x + 3) − y( x + 3) = 0
2
( x − y)( x + 3) = 0
2
x − y=0
2
x+3=0
y= x 2

30. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2
x ( x + 3) − y( x + 3) = 0
2
( x − y)( x + 3) = 0
2
x − y=0
2
x+3=0
y= x 2 x=3

31. Example 3
Describe the set of ordered pairs (x, y) satisfying
x + 3x − yx − 3 y = 0
3 2
( x + 3x ) + (− yx − 3 y) = 0
3 2
x ( x + 3) − y( x + 3) = 0
2
( x − y)( x + 3) = 0
2
x − y=0
2
x+3=0
y= x 2 x=3
The union of the sets of ordered pairs satisfying y = x2 and x = 3.

32. Homework

33. Homework
p. 610 #1-17