1. I am a problem solver Not a problem maker
I am an answer giver not an answer taker
I can do problems I haven’t seen
I know that strict doesn’t equal mean
In this room we do our best
Taking notes -practice –quiz- test
The more I try the better I feel
My favorite teacher is Mr. Meal…..ey

2. 1st block
3rd block
4th block

5. Take a closer look at the original equation and our roots:
x3 – 5x2 – 2x + 24 = 0
The roots therefore are: -2, 3, 4
What do you notice?
-2, 3, and 4 all go into the last term, 24!

6. Spooky! Let’s look at another
24x3 – 22x2 – 5x + 6 = 0
This equation factors to:
(x+1)(x-2)(x-3)= 0
The roots therefore are: -1/2, 2/3, 3/4

7. Take a closer look at the original equation and our roots:
24x3 – 22x2 – 5x + 6 = 0
This equation factors to:
(x+1)(x-2)(x-3)= 0
The roots therefore are: -1, 2, 3
What do you notice?
The numerators 1, 2, and 3 all go into the last term, 6!
The denominators (2, 3, and 4) all go into the first term, 24!

8. This leads us to the Rational Root Theorem
For a polynomial,
If p/q is a root of the polynomial,
then p is a factor of an
and q is a factor of ao

9. Example (RRT) ±3, ±1 ±1 ±12, ±6 , ±3 , ± 2 , ±1 ±4 ±1 , ±3
1. For polynomial
Here p = -3 and q = 1
Factors of -3
Factors of 1
±3, ±1 ±1 
Or 3,-3, 1, -1
Possible roots are ___________________________________
2. For polynomial
Here p = 12 and q = 3
Factors of 12
Factors of 3
±12, ±6 , ±3 , ± 2 , ±1 ±4
±1 , ±3 
Possible roots are ______________________________________________
Or ±12, ±4, ±6, ±2, ±3, ±1, ± 2/3, ±1/3, ±4/3
Wait a second Where did all of these come from???

10. Let’s look at our solutions
±12, ±6 , ±3 , ± 2 , ±1, ±4
±1 , ±3
Note that + 2 is listed twice; we only consider it as one answer
Note that + 1 is listed twice; we only consider it as one answer
Note that + 4 is listed twice; we only consider it as one answer
That is where our 9 possible answers come from!

11. Let’s Try One
Find the POSSIBLE roots of 5x3-24x2+41x-20=0

12. Let’s Try One
5x3-24x2+41x-20=0

13. Find all the possible rational roots

14. That’s a lot of answers!
Obviously 5x3-24x2+41x-20=0 does not have all of those roots as answers.
Remember: these are only POSSIBLE roots. We take these roots and figure out what answers actually WORK.

15. Step 1 – find p and q
p = -3
q = 1
Step 2 – by RRT, the only rational root is of the form…
Factors of p
Factors of q

16. Step 3 – factors
Factors of -3 = ±3, ±1
Factors of 1 = ± 1
Step 4 – possible roots
-3, 3, 1, and -1

17. Step 5 – Test each root X X³ + X² – 3x – 3 1 -3 3 1 -1
(-3)³ + (-3)² – 3(-3) – 3 = -12
(3)³ + (3)² – 3(3) – 3 = 24
(1)³ + (1)² – 3(1) – 3 = -4
(-1)³ + (-1)² – 3(-1) – 3 = 0
THIS IS YOUR ROOT
BECAUSE WE ARE LOOKING
FOR WHAT ROOTS WILL
MAKE THE EQUATION =0

19. Ex. 1: Factor a2 - 64 a2 – 64 = (a)2 – (8)2 = (a – 8)(a + 8)
You can use this rule to factor trinomials that can be written in the form a2 – b2.
a2 – 64 = (a)2 – (8)2
= (a – 8)(a + 8)

20. Ex. 2: Factor 9x2 – 100y2 9x2 – 100y2 = (3x)2 – (10y)2
You can use this rule to factor trinomials that can be written in the form a2 – b2.
9x2 – 100y2 = (3x)2 – (10y)2
= (3x – 10y)(3x + 10y)

21. a2 – 64 = (a – 8)(a + 8)
9x2 – 100y2 = (3x – 10y)(3x + 10y)

22. The sum or difference of two cubes will factor into a
binomial  trinomial.
same sign
always +
always opposite
same sign
always +
always opposite

25. Homework!!!

26. CONSTRUCTING THE TRIANGLE2
CONSTRUCTING THE TRIANGLE
ROW 0
ROW 1
ROW 2
ROW 3
R0W 4
ROW 5
ROW 6
ROW 7
ROW 8
ROW 9

27. Pascal’s Triangle and the Binomial Theorem
(x + y)0
= 1
(x + y)1
= 1x + 1y
(x + y)2
= 1x2 + 2xy + 1y2
(x + y)3
= 1x3 + 3x2y + 3xy2 +1 y3
(x + y)4
= 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
(x + y)5
= 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5
(x + y)6
= 1x6 + 6x5y1 + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + 1y6
7.5.2

28. Binomial Expansion - Practice
a = 3x
b = 2
Expand the following.
a) (3x + 2)4
= 4C0(3x)4(2)0
+ 4C1(3x)3(2)1
+ 4C2(3x)2(2)2
+ 4C3(3x)1(2)3
+ 4C4(3x)0(2)4
= 1(81x4)
+ 4(27x3)(2)
+ 6(9x2)(4)
+ 4(3x)(8)
+ 1(16)
= 81x x x2 + 96x +16
n = 4
a = 2x
b = -3y
b) (2x - 3y)4
= 4C0(2x)4(-3y)0
+ 4C1(2x)3(-3y)1
+ 4C2(2x)2(-3y)2
+ 4C3(2x)1(-3y)3
+ 4C4(2x)0(-3y)4
= 1(16x4)
+ 4(8x3)(-3y)
+ 6(4x2)(9y2)
+ 4(2x)(-27y3)
+ 81y4
= 16x4 - 96x3y + 216x2y xy3 + 81y4
7.5.6