The binomial theorem. Pascal’s Triangle.

Slides:

Advertisements

Similar presentations

Binomial Theorem 11.7.

Advertisements

Year 12 C1 Binomial Theorem. Task Expand the following: 1. (x + y) 1 2. (x + y) 2 3. (x + y) 3 4. (x + y) 4 What do you notice? Powers of x start from.

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.

Chapter 8 Sec 5 The Binomial Theorem. 2 of 15 Pre Calculus Ch 8.5 Essential Question How do you find the expansion of the binomial (x + y) n ? Key Vocabulary:

The Binomial Theorem.

Notes 9.2 – The Binomial Theorem. I. Alternate Notation A.) Permutations – None B.) Combinations -

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

2.4 Use the Binomial Theorem Test: Friday.

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!

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

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

5-7: The Binomial Theorem

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

Section 3-2 Multiplying Polynomials

2-6 Binomial Theorem Factorials

Pg. 601 Homework Pg. 606#1 – 6, 8, 11 – 16 # … + (2n) 2 # (3n + 1) #5 #7(3n 2 + 7n)/2 #84n – n 2 #21#23 #26 #29 #33The series.

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

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

5.4 Binomial Coefficients Theorem 1: The binomial theorem Let x and y be variables, and let n be a nonnegative integer. Then Example 3: What is the coefficient.

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.

Warm-up Time-limit….. 5 P 2 = 11 ! = 7 C 3 = 7!___ = 3!(7-3)! 5 C C 3 =

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.

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.

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.

Objective: To use Pascal’s Triangle and to explore the Binomial Theorem.

Binomial Theorem and Pascal’s Triangle.

Homework Questions? Daily Questions: How do I Multiply Polynomials?

3-2 Multiplying polynomials

The Binomial Theorem.

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

4.2 Pascal’s Triangle and the Binomial Theorem

Pascals Triangle and the Binomial Expansion

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.

Binomial Expansion.

Objectives Multiply polynomials.

MATH 2160 Pascal’s Triangle.

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

Use Pascal’s triangle to expand the expression (3 x - 2 y) 3

11.9 Pascal’s Triangle.

How do I Multiply Polynomials? How do I use Binomial Expansion?

Homework Questions? Daily Questions: How do I Multiply Polynomials?

11.6 Binomial Theorem & Binomial Expansion

Binomial Theorem; Pascal’s Triangle

The Binomial Theorem OBJECTIVES: Evaluate a Binomial Coefficient

Chapter 12 Section 4.

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

Unit 5 Polynomial Operations

Pascal’s Triangle.

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

Presentation transcript:

The binomial theorem. Pascal’s Triangle. What you’ll learn To expand a binomial using Pascal’s Triangle. To use the binomial theorem. Vocabulary Expand, Pascal’s triangle, binomial theorem.

Essential understanding: Expand using the distributive property. Row Power Expand form Coefficients only

And this is Pascal’s triangle: this is a triangular array of number in which the first and last number of each row is 1. Example 1: Using the Pascal’s triangle. What is the expansion of ? The exponents for a begin with 6 and decrease to 0 The exponents for b begin with o and increase to 6

Binomial theorem: For every positive integer n, Your turn: What is the expansion of (a + b)^8? Answer

Your turn again What is the expansion of (3x – 2)^5? Use the binomial theorem. Think

Classwork and homework From the text book pages 329-333 Exercises 8 to 53