10.2b - Binomial Theorem.

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.
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.
Ms. Nong Digital Lesson (Play the presentation and turn on your volume)
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.
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)
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 ©
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.
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.
Binomial Theorem & Binomial Expansion
The Binomial Theorem. (x + y) 0 Find the patterns: 1 (x + y) 1 x + y (x + y) 2 (x + y) 3 x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 (x + y) 0 (x + y) 1 (x +
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.
(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????
2-6 Binomial Theorem Factorials
How many different landscapes could be created?
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
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.
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.
APC Unit 2 CH-12.5 Binomial Theorem. Warm-up  Take your Homework out  Clearly Label 12.2, 12.3, and 12.4  Ask your Questions  While I’m Checking…
Combination
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.
Splash Screen.
Splash Screen.
The binomial expansions
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
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
Binomial Expansion.
The Binomial Theorem.
6.8 – Pascal’s Triangle and the Binomial Theorem
Ch 4.2: Adding, Subtracting, and Multiplying 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.
Use the Binomial Theorem
Use Pascal’s triangle to expand the expression (3 x - 2 y) 3
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)
Digital Lesson The Binomial Theorem.
6.8 – Pascal’s Triangle and the Binomial Theorem
HW: Finish HPC Benchmark 1 Review
ALGEBRA II HONORS/GIFTED - SECTION 5-7 (The Binomial Theorem)
The Binomial Theorem.
Pascal’s Triangle.
10.4 – Pascal’s Triangle and the Binomial Theorem
Warm Up 1. 10C P4 12C P3 10C P3 8C P5.
Presentation transcript:

10.2b - Binomial Theorem

Consider the expansion: What are some different patterns you see in the expansion above?

What are some different patterns you see? # of terms in expansion is one more than the binomials power 2. Powers of x go from n0; powers of y from 0n 3. 1st and last coefficients is one 4. Term # is always one more than the power of y

Each row is the coefficients for n = row # Fill in the coefficients of the (above) expansions below: Row:#___ n = 0 1 n = 1 Row:#___ 1 1 1 1 2 1 Row:#___ 2 n = 2 1 3 3 1 Row:#___ 3 n = 3 1 4 6 4 1 Row:#___ 4 n = 4 1 5 10 10 5 1 n = 5 Row:#___ 5 n = 6 1 6 15 20 15 6 1 Row:#___ 6 Each row is the coefficients for n = row #

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Pascal’s Triangle This is called: ___________ _________ The ‘math stud’ who is given credit for this is: _____________________________ Blaise Pascal (1623 – 1662)

Binomial Theorem The binomial expansion of Where the coefficients of each term are given by: n = power of binomial m = the “y” exponent

1. Expand the binomial: Find the coefficients using Pascal’s triangle Label the same coefficients using the binomial theorem a) b)

2. Find the binomial coefficients of the given terms using the binomial theorem (check your answers with the answers you got in #1) 3 15 a) 3 15 b)

6 2 2 20 c) y3 d) The fourth term of the binomial: 2. Find the binomial coefficients of the given terms using the binomial theorem (check your answers with the answers you got in #1) c) 6 y3 d) The fourth term of the binomial: 2 2 20

2. Find the binomial coefficients of the given terms using the binomial theorem (check your answers with the answers you got in #1) e) 1

3. Expand the binomial: Hint: a) write out: (x + y)5 b) substitute _____ in for y c) simplify each term -2

Coef x y Term (Pascal) 1 x5 (-2)0 = 1 x5 5 x4 (-2)1 = -2 -10x4 10 x3 (-2)2 = 4 40x3 x2 (-2)3 = -8 10 -80x2 x1 (-2)4 = 16 80x1 5 x0 = 1 (-2)5 = -32 -32 1

3 4 84 4. Find the fourth term of the binomial: Coef x y Term (binom) (-1)3 = -1 -84m6 x9 x8y x7y2 x6y3 3 4 84

5. Expand the binomial: Hint: a) write out: (2a + 3b)4 b) substitute _____ for x & _____ for y c) simplify each term 2a 3b

Coef x y Term (Pascal) 1 (2a)4 = 16a4 (3b)0 = 1 16a4 4 (2a)3 = 8a3 (3b)1 = 3b 96a3b 6 (2a)2 = 4a2 (3b)2 = 9b2 216a2b2 4 (2a)1 = 2a (3b)3 = 27b3 216ab3 1 (2a)0 = 1 (3b)4 = 81b4 81b4