(a + b) 0 =1 (a + b) 1 = (a + b) 2 = (a + b) 3 = 1a 1 + 1b 1 1a 2 + 2ab + 1b 2 1a 3 + 3a 2 b + 3ab 2 + 1b 3 Binomial Expansion... What do we notice????

Slides:

Advertisements

Similar presentations

6.8 – Pascal’s Triangle and the Binomial Theorem.

Advertisements

The binomial theorem 1 Objectives: Pascal’s triangle Coefficient of (x + y) n when n is large Notation: ncrncr.

Sequences, Induction and Probability

Math 143 Section 8.5 Binomial Theorem. (a + b) 2 =a 2 + 2ab + b 2 (a + b) 3 =a 3 + 3a 2 b + 3ab 2 + b 3 (a + b) 4 =a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b.

Monday: Announcements Progress Reports this Thursday 3 rd period Tuesday/Wednesday STARR Testing, so NO Tutorials (30 minute classes) Tuesday Periods 1,3,5,7.

The Binomial Theorem.

Copyright © 2011 Pearson, Inc. 9.2 The Binomial Theorem.

What does Factorial mean? For example, what is 5 factorial (5!)?

Binomial Expansion Honors Advanced Algebra Presentation 2-3.

BINOMIAL EXPANSION. Binomial Expansions Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 The binomial theorem provides a useful method.

13.5 The Binomial Theorem. There are several theorems and strategies that allow us to expand binomials raised to powers such as (x + y) 4 or (2x – 5y)

11-10 The Binomial Formula  Recall: (a + b) 2 = a 2 +2ab + b 2  For any binomial expansion: (a + b) n There are (n + 1) terms. Each term is n th degree.

11.1 – Pascal’s Triangle and the Binomial Theorem

Warm up 1. Write the expression in expanded form, then find the sum. 2. Express the series using sigma notation.

Copyright © Cengage Learning. All rights reserved. 8.4 The Binomial Theorem.

9.5 The Binomial Theorem Let’s look at the expansion of (x + y)n

Binomial – two terms Expand (a + b) 2 (a + b) 3 (a + b) 4 Study each answer. Is there a pattern that we can use to simplify our expressions?

The Binomial Theorem.

Copyright © Cengage Learning. All rights reserved. 8 Sequences, Series, and Probability.

Binomial Theorem & Binomial Expansion

Pascal’s Triangle and the Binomial Theorem (x + y) 0 = 1 (x + y) 1 = 1x + 1y (x + y) 2 = 1x 2 + 2xy + 1y 2 (x + y) 3 = 1x 3 + 3x 2 y + 3xy 2 +1 y 3 (x.

Binomial Theorem Binomial Theorem Term 1 : Unit 3

2-6 Binomial Theorem Factorials

How many different landscapes could be created?

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Pg. 606 Homework Pg. 606 #11 – 20, 34 #1 1, 8, 28, 56, 70, 56, 28, 8, 1 #2 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1 #3 a5 + 5a4b + 10a3b2 + 10a2b3.

8.5 The Binomial Theorem. Warm-up Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3.

The Binomial Theorem Section 9.2a!!!. Powers of Binomials Let’s expand (a + b) for n = 0, 1, 2, 3, 4, and 5: n Do you see a pattern with the binomial.

Section 8.5 The Binomial Theorem. In this section you will learn two techniques for expanding a binomial when raised to a power. The first method is called.

Section 8.5 The Binomial Theorem.

Factorials and Sequences

Binomial Theorem and Pascal’s Triangle.

Sequences, Series, and Probability

The binomial expansions

7B and 7C This lesson is for Chapter 7 Section B

Pascal’s Triangle and the Binomial Theorem

Copyright © Cengage Learning. All rights reserved.

Section 9-5 The Binomial Theorem.

The Binomial Theorem; Pascal’s Triangle

9.5 The Binomial Theorem Let’s look at the expansion of (x + y)n

10.2b - Binomial Theorem.

Binomial Expansion.

Digital Lesson The Binomial Theorem.

The Binomial Theorem.

8.4 – Pascal’s Triangle and the Binomial Theorem

Digital Lesson The Binomial Theorem.

TCM – DO NOW A peer believes that (a+b)3 is a3 + b3 How can you convince him that he is incorrect? Write your explanation in your notes notebook.

Binomial Theorem Pascal’s Triangle

Binomial Theorem Introduction

11.6 Binomial Theorem & Binomial Expansion

LESSON 10–5 The Binomial Theorem.

Digital Lesson The Binomial Theorem.

Chapter 12 Section 4.

Digital Lesson The Binomial Theorem.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

ALGEBRA II HONORS/GIFTED - SECTION 5-7 (The Binomial Theorem)

Digital Lesson The Binomial Theorem.

Digital Lesson The Binomial Theorem.

HW: Finish HPC Benchmark 1 Review

ALGEBRA II HONORS/GIFTED - SECTION 5-7 (The Binomial Theorem)

The Binomial Theorem.

9.2 The Binomial Theorem.

10.4 – Pascal’s Triangle and the Binomial Theorem

Warm Up 1. 10C P4 12C P3 10C P3 8C P5.

Section 11.7 The Binomial Theorem

pencil, red pen, highlighter, GP notebook, calculator

Presentation transcript:

(a + b) 0 =1 (a + b) 1 = (a + b) 2 = (a + b) 3 = 1a 1 + 1b 1 1a 2 + 2ab + 1b 2 1a 3 + 3a 2 b + 3ab 2 + 1b 3 Binomial Expansion... What do we notice????

Pascal’s Triangle

Example Expand (a + b) 6 Solution: (a + b) 6 = a 6 + 6a 5 b + 15a 4 b a 3 b a 2 b 4 + 6ab 5 + b 6

Example Find the first four terms in the expansion of (x – 2y) 7. Solution (a + b) 7 = a 7 + 7a 6 b + 21a 5 b a 4 b 3 + ….. Substitute: x = a, 2y = b (x + 2y) 7 = x 7 + 7x 6 (2y) + 21x 5 (2y) x 4 (2y) Simplify (x + 2y) 7 = x x 6 y + 84x 5 y x 4 y 3

Example a a 11 b + 66a 10 b a 9 b 3

Factorial Notation r! = r(r-1)(r-2)(r-3)…..3*2*1 9! = 9*8*7*6*5*4*3*2*1 = ! = 12*11*10*9*8*7*6*5*4*3*2*1 = What happens when we divide factorials? Calculators can help! MATH -> PRB -> #4

Therefore, you can write the fourth term of the expansion of (a + b) 12 as follows: Please write a formula for the k term of the expansion of (a + b) n.

Example Find and simplify the seventh term in the expansion of (2x-y) (2x) 4 (-y) 6 = 210 * 16x 4 * y 6 = 3360x 4 y 6