Download presentation

Presentation is loading. Please wait.

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 TRIANGLE**

2 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

Similar presentations

Presentation is loading. Please wait....

OK

Least Common Multiples and Greatest Common Factors

Least Common Multiples and Greatest Common Factors

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on single phase and three phase dual converter circuit Simple ppt on nuclear power plant Pdf to ppt online converter free Raster scan display ppt on tv Ppt on cctv camera technology Ppt on france in french language Ppt online marketing Ppt on memory management in operating system Ppt on importance of sports in our life Ppt on word association test example