How many different landscapes could be created?

Slides:



Advertisements
Similar presentations
Digital Lesson The Binomial Theorem.
Advertisements

Expansion of Binomials. (x+y) n The expansion of a binomial follows a predictable pattern Learn the pattern and you can expand any binomial.
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.
Warm Up Multiply. 1. x(x3) x4 2. 3x2(x5) 3x7 3. 2(5x3) 10x3 4. x(6x2)
SFM Productions Presents: Another adventure in your Pre-Calculus experience! 9.5The Binomial Theorem.
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!)?
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 ©
Copyright © 2007 Pearson Education, Inc. Slide 8-1.
11.1 – Pascal’s Triangle and the Binomial Theorem
Aim: Binomial Theorem Course: Alg. 2 & Trig. Do Now: Aim: What is the Binomial Theorem and how is it useful? Expand (x + 3) 4.
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????
Binomial Theorem Binomial Theorem Term 1 : Unit 3
2-6 Binomial Theorem Factorials
Do Now: Evaluate each expression for x = -2. Aim: How do we work with polynomials? 1) -x + 12) x ) -(x – 6) Simplify each expression. 4) (x + 5)
A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Essential Questions How do we multiply polynomials?
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.
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.
Binomial Theorem and Pascal’s Triangle.
Splash Screen.
The Binomial & Multinomial Coefficients
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
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.
Binomial Expansion.
Digital Lesson The Binomial Theorem.
The Binomial Theorem.
Objectives Multiply polynomials.
8.4 – Pascal’s Triangle and the Binomial Theorem
Digital Lesson The Binomial Theorem.
MATH 2160 Pascal’s Triangle.
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
11.9 Pascal’s Triangle.
11.6 Binomial Theorem & Binomial Expansion
Digital Lesson The Binomial Theorem.
The Binomial Theorem OBJECTIVES: Evaluate a Binomial Coefficient
Digital Lesson The Binomial Theorem.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Digital Lesson The Binomial Theorem.
Digital Lesson The Binomial Theorem.
HW: Finish HPC Benchmark 1 Review
The Binomial Theorem.
10.4 – Pascal’s Triangle and the Binomial Theorem
pencil, red pen, highlighter, GP notebook, calculator
Do Now: Aim: How do we work with polynomials?
Presentation transcript:

How many different landscapes could be created? Myriorama cards were invented in France around 1823 by Jean-Pierre Brès and further developed in England by John Clark. Early myrioramas were decorated with people, buildings, and scenery that could be laid out in any order to create a variety of landscapes. One 24-card set is sold as “The Endless Landscape.” How many different landscapes could be created? Math 30-1

11.3 The Binomial Theorem Student Activity Race Expand (x + y)0 Math 30-1

Pascal’s Triangle and the Binomial Theorem (x + y)n, n ∈ N (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 What is the relationship between the value of n and the number of terms in the expansion? Math 30-1

Pascal’s Triangle and the Binomial Theorem The numerical coefficients in a binomial expansion can be found in Pascal’s triangle. Pascal’s Triangle Pascal’s Triangle using Combinatorics n = 0 (a + b)0 1st Row 0C0 1 n = 1 (a + b)1 2nd Row 1C0 1C1 1 1 3rd Row n = 2 (a + b)2 1 2 1 2C0 2C1 2C2 n = 3 (a + b)3 4th Row 1 3 3 1 3C0 3C1 3C2 3C3 n = 4 (a + b)4 5th Row 1 4 6 4 1 4C0 4C1 4C2 4C3 4C4 1 5 1 n = 5 (a + b)5 5 10 10 5C0 5C1 5C2 5C3 5C4 5C5 6th Row What patterns do you notice in Pascal’s Triangle? Math 30-1

The Binomial Theorem The Binomial Theorem is a formula used for expanding powers of binomials. (a + b)3 = (a + b)(a + b)(a + b) = a3 + 3a2b + 3ab2 + b3 The first term has no b. It is like choosing no b from three b’s. The combination 3C0 is the coefficient of the first term. The second term has one b. It is like choosing one b from three b’s. The combination 3C1 is the coefficient of the second term. The third term has two b’s. It is like choosing two b’s from three b’s. The combination 3C2 is the coefficient of the third term. The fourth term has three b’s. It is like choosing three b’s from three b’s. The combination 3C3 is the coefficient of the fourth term. (a + b)3 = 3C0 a3b0 + 3C1 a2b1 + 3C2 ab2 + 3C3 b3 Math 30-1

Binomial Expansion - the General Term (a + b)3 = a3 + 3a2b + 3ab2 + b3 The degree of each term is 3. For the variable a, the degree descends from 3 to 0. For the variable b, the degree ascends from 0 to 3. (a + b)3 = 3C0 a3 - 0b0 + 3C1 a3 - 1b1 + 3C2 a3 - 2b2 + 3C3 a3 - 3b3 (a + b)n = nC0 an - 0b0 + nC1 an - 1b1 + nC2 an - 2b2 + …..+ nCk an - kbk The general term is the (k + 1)th term: tk + 1 = nCk xn - k yk The number of terms in the expansion of (a + b)n is n + 1 Math 30-1

How many terms are in the expansion of each binomial? If there are 17 terms in the expansion of the given binomial, what is the value of k? Math 30-1

Binomial Expansion - Practice x = 3x y = 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) = 81x4 + 216x3 + 216x2 + 96x + 16 n = 4 x = 2x y = -3y b) (2x - 3y)3 = 3C0(2x)3(-3y)0 + 3C1(2x)2(-3y)1 + 3C2(2x)1(-3y)2 + 3C3(2x)0(-3y)3 = 1(8x3)(1) + 3(4x2)(-3y) + 3(2x)(9y2) + 1(1)(-27y3) = 8x3 - 36x2y + 24x1y2 - 2y3 Math 30-1

Binomial Expansion - Practice (2x - 3x -1)5 = 5C0(2x)5(-3x -1)0 + 5C1(2x)4(-3x -1)1 + 5C2(2x)3(-3x -1)2 n = 5 x = 2x y = -3x -1 + 5C3(2x)2(-3x -1)3 + 5C4(2x)1(-3x -1)4 + 5C5(2x)0(-3x -1)5 = 1(32x5) + 5(16x4)(-3x -1) + 10(8x3)(9x -2) + 10(4x2)(-27x -3) + 5(2x)(81x -4) + 1(-243x -5) = 32x5 - 240x3 + 720x1 - 1080x -1 + 810x -3 - 243x -5 Math 30-1

Finding a Particular Term in a Binomial Expansion a) Determine an expression for the eighth term in the expansion of (3x - 2)11. tk + 1 = nCk xn - k yk t8 = t7 + 1 so k = 7 n = 11 x = 3x y = -2 k = 7 t7 + 1 = 11C7 (3x)11 - 7 (-2)7 t8 = 11C7 (3x)4 (-2)7 = 330(81x4)(-128) = -3 421 440 x4 Determine the coefficient of the term containing Ax2y5 in the expansion of(x - 3y)7. tk + 1 = nCk xn - k yk n = 7 k = 5 Ax2y5 = 7C5 (x)2(-3y)5 = 21 (x2)(-243y5) = -5103x2y5 Math 30-1

Determine an expression for the term containing x 3 in the expansion of (2x - 3)6. tk + 1 = nCk xn - k yk When the exponent on x is 3, and the value of n is 6, the value of k must be 3. t3 + 1 = 6C3 (2x)6 - 3 (-3)3 Math 30-1

Determine the value of the constant term of the expansion of tk + 1 = nCk xn - k yk x18 - 3k = x0 n = 18 x = 2x y = -x -2 k = ? tk + 1 = 18Ck (2x)18 - k (-x -2)k 18 - 3k = 0 -3k = -18 k = 6 tk + 1 = 18Ck 218 - k x18 - k (-1)k x -2k tk + 1 = 18Ck 218 - k (-1)k x18 - k x -2k tk + 1 = 18Ck 218 - k (-1)k x18 - 3k Substitute k = 6: t6 + 1 = 18C6 218 - 6 (-1)6 x18 - 3(6) Therefore, the constant term is 76 038 144. t7 = 18C6 212 (-1)6 t7 = 76 038 144 Math 30-1

Determine a Particular Term in the Binomial One term in the expansion of (2x - m)7 is -15 120x4y3. Determine an expression to represent m. tk + 1 = nCk an - k bk n = 7 a = 2x b = -m k = 3 tk + 1 = 7C3 (2x)4 (-m)3 tk + 1 = (35) (16x4) (-m)3 -15 120x4y3 = (560x4) (-m)3 Therefore, m is 3y. 3y = m Math 30-1

Assignment Worksheet Math 30-1

Pascal’s triangle is an array of natural numbers. The sum of any two adjacent numbers is equal to the number directly below them. Sum of each row 0 Row 1 1 20 1st Row 1 1 2 21 2nd Row 1 2 1 4 22 3rd Row 1 3 3 1 8 23 4th Row 1 4 6 4 1 16 24 1 32 25 5th Row 1 5 10 10 5 26 1 6 15 20 15 6 1 64 6th Row 1 7 21 35 35 21 7 1 128 27 7th Row nth Row 2n Math 30-1