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

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.

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)

The Binomial Theorem.

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.

Multiplying Polynomials

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????

Section 3-2 Multiplying Polynomials

2-6 Binomial Theorem Factorials

Holt McDougal Algebra Multiplying Polynomials Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive.

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.

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.

3-2 Multiplying polynomials

Sequences, Series, and Probability

The binomial expansions

Section 9-5 The Binomial Theorem.

Use the Binomial Theorem

The Binomial Theorem Ms.M.M.

The Binomial Expansion Chapter 7

4.2 Pascal’s Triangle and the Binomial Theorem

The Binomial Theorem; Pascal’s Triangle

Use the Binomial Theorem

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

10.2b - Binomial Theorem.

Binomial Expansion.

The Binomial Theorem.

6.8 – Pascal’s Triangle and the Binomial Theorem

Objectives Multiply polynomials.

Multiplying Polynomials

Binomial Theorem Honor’s Algebra II.

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

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

LESSON 10–5 The Binomial Theorem.

Chapter 12 Section 4.

The binomial theorem. Pascal’s Triangle.

Multiplying Polynomials

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

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)

9.2 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:

ALGEBRA II HONORS/GIFTED - SECTION 5-7 (The Binomial Theorem) @ SECTION 5-7 : THE BINOMIAL THEOREM

Work out the problem that corresponds to your group number. 9) What do you notice about the signs before each term? (x + y)0 (x – y)0 (x + y)1 (x – y)1 (x + y)2 (x – y)2 (x + y)3 8) (x – y)3 = 1 = x + y = x – y = x2 + 2xy + y2 = x2 – 2xy + y2 = x3 + 3x2y + 3xy2 + y3 = x3 – 3x2y + 3xy2 – y3 10) And, those exponents. What about them? 11) Hey, what’s going on with the coefficients?

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 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 This is Pascal’s Triangle. The triangle’s row are numbered beginning with row 0.

Expand. a) How many terms are in the expansion? 9a) (x + y)4 ANSWER : 5 b) How are the signs of the terms arranged? + (1) (4) (6) (x)4 (x)3 (x)2 (x)1 (x)0 (y)0 (y)1 (y)2 (y)3 (y)4 ANSWER : all + c) Lay down the coefficients from Pascal’s Triangle. d) Place the terms of the binomial down noting their exponents. e) Simplify. = x4 + 4x3y + 6x2y2 + 4xy3 + y4

f) (x – y)5 ANSWER : x5 - 5x4y + 10x3y2 - 10x2y3 + 5xy4 - y5 g) (x – y)6 ANSWER : x6 – 6x5y + 15x4y2 – 20x3y3 + 15x2y4 – 6xy5 + y6 Find the specified term. 10a) Third term of (x + 2y)6 ANSWER : 60x4y2 b) Fourth term of (2x2 – y)5 ANSWER : -40x4y3

11)a) Expand using Pascal’s Triangle : (x – 4)5 ANSWER : x5 – 20x4 + 160x3 – 640x2 + 1280x - 1024 b) Now, find : 5C0, 5C1, 5C2, 5C3, 5C4, and 5C5. ANSWERS : 1, 5, 10, 10, 5, 1 c) How do these answers compare with the coefficients found in Pascal’s Triangle? ANSWER : They are the same.

12) Expand. SOLUTION : +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 a) (3x – 2)4 = 81x4 – 216x3 + 216x2 – 96x + 16

b) (4x + y2)3 ANSWER : 64x3 + 48x2y2 + 12xy4 + y6 c) (a + a-1)6 ANSWER : a6 + 6a4 + 15a2 + 20 + 15a-2 + 6a-4 + a-6 d) (1 – i)5 ANSWER : -4 + 4i

13) Find the specified term. Must add to 9. a) (x + y)9 ; y4 SOLUTION : 9C4 (x)5 (y)4 =126x5y4 Where does the “9” come from? These numbers are always the same. b) (2x – 3y)7 ; y4 ANSWER : 7C4 (2x)3 (-3y)4 = 22680x3y4

c) (a – 2b)8 ; a3 ANSWER : 8C5 (a)3 (-2b)5 = -1792a3b5 d) (a + b)n ; br ANSWER : nCr (a)n-r (b)r

14) Find the specified term. a) (m + n)20 ; 12th term SOLUTION : 20C11 (m)9 (n)11 = 167960m9n11 The choose number is always 1 less than the term number. The choose number is also known as the “magic number”. b) (r3 – 2s)9 ; 5th term ANSWER : 9C4 (r3)5 (-2s)4 = 2016r15s4

c) (2x – y)12 ; middle term ANSWER : 12C6 (2x)6 (-y)6 = 59136x6y6 d) (a + 2b3)10 ; middle term ANSWER : 10C5 (a)5 (2b)5 = 8064a5b5