Binomial Expansion.

Slides:

Advertisements

Similar presentations

Binomial Theorem 11.7.

Advertisements

6.8 – Pascal’s Triangle and the Binomial Theorem.

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

Warm Up Multiply. 1. x(x3) x4 2. 3x2(x5) 3x7 3. 2(5x3) 10x3 4. x(6x2)

Ms. Nong Digital Lesson (Play the presentation and turn on your volume)

The Binomial Theorem.

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

2.4 Use the Binomial Theorem Test: Friday.

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)

Pascal’s Triangle and the Binomial Theorem, then Exam!

The Binomial Theorem 9-5. Combinations How many combinations can be created choosing r items from n choices. 4! = (4)(3)(2)(1) = 24 0! = 1 Copyright ©

Copyright © 2007 Pearson Education, Inc. Slide 8-1.

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.

Lesson 6.8A: The Binomial Theorem OBJECTIVES:  To evaluate a binomial coefficient  To expand a binomial raised to a power.

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?

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.

2-6 Binomial Theorem Factorials

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

The Binomial Theorem (a+b)2 (a+b)(a+b) a2 +2ab + b2

7.1 Pascal’s Triangle and Binomial Theorem 3/18/2013.

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.

Essential Questions How do we multiply polynomials?

Algebra 2 CC 1.3 Apply the Binomial Expansion Theorem Recall: A binomial takes the form; (a+b) Complete the table by expanding each power of a binomial.

By: Michelle Green. Construction of Pascal’s Triangle ROW 0 ROW 1 ROW 2 ROW 3 ROW 4 ROW ROW 6 ALWAYS.

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.

Binomial Theorem and Pascal’s Triangle.

The binomial expansions

Pascal’s Triangle and the Binomial Theorem

Section 9-5 The Binomial Theorem.

Use the Binomial Theorem

The Binomial Theorem Ms.M.M.

The Binomial Expansion Chapter 7

6-8 The Binomial Theorem.

A quick and efficient way to expand binomials

4.2 Pascal’s Triangle and the Binomial Theorem

The Binomial Theorem; Pascal’s Triangle

Use the Binomial Theorem

The Binomial Theorem Objectives: Evaluate a Binomial Coefficient

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

10.2b - Binomial Theorem.

The Binomial Theorem.

6.8 – Pascal’s Triangle and the Binomial Theorem

Objectives Multiply polynomials.

8.4 – Pascal’s Triangle and 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

4-2 The Binomial Theorem Use Pascal’s Triangle to expand powers of binomials Use the Binomial Theorem to expand powers of binomials.

Essential Questions How do we use the Binomial Theorem to expand a binomial raised to a power? How do we find binomial probabilities and test hypotheses?

Use the Binomial Theorem

Binomial Expansion L.O. All pupils understand why binomial expansion is important All pupils understand the pattern binomial expansion follows All pupils.

11.9 Pascal’s Triangle.

11.6 Binomial Theorem & Binomial Expansion

The Binomial Theorem OBJECTIVES: Evaluate a Binomial Coefficient

©2001 by R. Villar All Rights Reserved

The binomial theorem. Pascal’s Triangle.

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

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

6.8 – Pascal’s Triangle and the Binomial Theorem

HW: Finish HPC Benchmark 1 Review

Unit 5 Polynomial Operations

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

10.4 – Pascal’s Triangle and the Binomial Theorem

Two hungry Math 160 students are waiting in line at Chipotle

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

Presentation transcript:

Binomial Expansion

The Binomial Theorem Strategy only: how do we expand these? 1. (x + 2)2 2. (2x + 3)2 3. (x – 3)3 4. (a + b)4

The Binomial Theorem Solutions 1. (x + 2)2 = x2 + 2x + 2x + 22 = x2 + 4x + 4 2. (2x + 3)2 = (2x)2 + (3)(2x) + (3)(2x) + 32 = 4x2 + 12x + 9 3. (x – 3)3 = (x – 3)(x – 3)2 = (x – 3)(x2 – 2(3)x + 32) = (x – 3)(x2 – 6x + 9) = x(x2 – 6x + 9) – 3(x2 – 6x + 9) = x3 – 6x2 + 9x – 3x2 + 18x – 27 = x3 – 9x2 + 27x – 27 4. (a + b)4 = (a + b)2(a + b)2 = (a2 + 2ab + b2)(a2 + 2ab + b2) = a2(a2 + 2ab + b2) + 2ab(a2 + 2ab + b2) + b2(a2 + 2ab + b2) = a4 + 2a3b + a2b2 + 2a3b + 4a2b2 + 2ab3 + a2b2 + 2ab3 + b4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

Isn’t there an easier way? THAT is a LOT of work! Isn’t there an easier way?

Introducing: Pascal’s Triangle Take a moment to copy the first 6 rows. What patterns do you see? Row 5 Row 6

The Binomial Theorem Use Pascal’s Triangle to expand (a + b)5. Use the row that has 5 as its second number. The exponents for a begin with 5 and decrease. 1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5 The exponents for b begin with 0 and increase. In its simplest form, the expansion is a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5. Row 5

The Binomial Theorem Use Pascal’s Triangle to expand (x – 3)4. First write the pattern for raising a binomial to the fourth power. 1 4 6 4 1 Coefficients from Pascal’s Triangle. (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 Since (x – 3)4 = (x + (–3))4, substitute x for a and –3 for b. (x + (–3))4 = x4 + 4x3(–3) + 6x2(–3)2 + 4x(–3)3 + (–3)4 = x4 – 12x3 + 54x2 – 108x + 81 The expansion of (x – 3)4 is x4 – 12x3 + 54x2 – 108x + 81.

Let’s Try Some Expand the following (3x-2y)4

Let’s Try Some Expand the following (3x-2y)4