The Binomial Theorem

Jul 11, 2012

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The Binomial Theorem

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1. The Binomial Theorem

The Binomial Theorem

The Binomial Theorem. Lecture 34 Section 6.7 Wed, Mar 28, 2007. The Binomial Theorem. Theorem: Given any numbers a and b and any nonnegative integer n , . The Binomial Theorem. Proof: Use induction on n . Base case: Let n = 0. Then ( a + b ) 0 = 1 and

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The Binomial Theorem

The Binomial Theorem. Macon State College Gaston Brouwer, Ph.D. February 2010. Binomial Theorem . Combinatorics Pascal’s Triangle Binomial Theorem Polynomials Applications Approximations Volume of a cube Questions. Combinatorics: Example 1.

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The Binomial Theorem

Precalculus. Lesson 9.2. The Binomial Theorem. Quick Review. What you’ll learn about. Powers of Binomials Pascal’s Triangle The Binomial Theorem Factorial Identities … and why The Binomial Theorem is a marvelous study in combinatorial patterns. Binomial Coefficient.

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The Binomial Theorem

Chapter 7 Combinatorics. 7.5. The Binomial Theorem. 7.5. 1. MATHPOWER TM 12, WESTERN EDITION. Pascal’s Triangle and the Binomial Theorem. ( x + y ) 0. = 1. ( x + y ) 1. = 1 x + 1 y. = 1 x 2 + 2 xy + 1 y 2. ( x + y ) 2. = 1 x 3 + 3 x 2 y + 3 xy 2 + 1 y 3.

382 views • 12 slides

The Binomial Theorem

The Binomial Theorem. 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 that without lengthy multiplication. Can you see a pattern? Can you make a guess what the next one would be?.

550 views • 11 slides

The Binomial Theorem

Chapter 7 Combinatorics. 7.5. The Binomial Theorem. 7.5. 1. MATHPOWER TM 12, WESTERN EDITION. Pascal’s Triangle and the Binomial Theorem. ( x + y ) 0. ( x + y ) 1. ( x + y ) 2. ( x + y ) 3. ( x + y ) 4. ( x + y ) 5. ( x + y ) 6. 7.5. 2. The Binomial Theorem.

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Binomial Theorem

2. Binomial Theorem. Case Study. 2.1 Summation Notation. 2.2 Binomial Theorem. Chapter Summary. Is it possible to find the value of (1.01) n by considering the expansion of (1  0.01) n step by step without using a calculator?.

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The Binomial Theorem

The Binomial Theorem. 11.4. Derive the binomial theorem Use the binomial theorem Apply Pascal’s triangle. Binomial Theorem. Expanding expressions in the form ( a + b ) n , where n is a natural number.

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The Binomial Theorem

The Binomial Theorem. Questions:. Put (x+y) 2 in standard form Put (x+y) 3 in standard form Put (x+y) 4 in standard form. Pascal’s Triangle. is a triangle of numbers, each new number being the sum of the two above it.

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The Binomial Theorem

The Binomial Theorem. ALGEBRA 2 LESSON 6-8. (For help, go to Lessons 5-1 and 6-7.). Multiply. 1. ( x + 2) 2 2. (2 x + 3) 2 3. ( x – 3) 3 4. ( a + b ) 4 Evaluate. 5. 5 C 0 6. 5 C 1 7. 5 C 2 8. 5 C 3 9. 5 C 4. 6-8. Solutions.

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The Binomial Theorem

The Binomial Theorem. Patterns in Binomial Expansions. By studying the expanded form of each binomial expression, we are able to discover the following patterns in the resulting polynomials. 1. The first term is a n . The exponent on a decreases by 1 in each successive term.

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The Binomial Theorem

The Binomial Theorem. Pascal’s Triangle and the Binomial Theorem. ( x + y ) 0. = 1. ( x + y ) 1. = 1 x + 1 y. = 1 x 2 + 2 xy + 1 y 2. ( x + y ) 2. = 1 x 3 + 3 x 2 y + 3 xy 2 + 1 y 3. ( x + y ) 3. ( x + y ) 4.

616 views • 12 slides

The Binomial Theorem

The Binomial Theorem. What about the coefficients of each term? Is there a pattern there?. Find the patterns:. 1. (x + y) 0. (x + y) 0. x 1 + y 1. x + y. (x + y) 1. (x + y) 1. x 2 + 2xy + y 2. (x + y) 2. (x + y) 2. (x + y) 3. (x + y) 3.

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The Binomial Theorem

The Binomial Theorem. Connecting binomial expansion to combinations. Used in science, probability theory, game theory, statistics and combinatorics. Expand (3x – 2) 10. (3x – 2)(3x – 2)(3x – 2)(3x – 2)(3x – 2)(3x –2)(3x – 2) (3x – 2)(3x – 2)(3x – 2) = Have fun!!!

442 views • 9 slides

Binomial Theorem

Binomial Theorem. 11.7. There are n+1 terms Functions of n Exponent of a in first term Exponent of b in last term Other terms Exponent of a decreases by 1 Exponent of b increases by 1. Sum of exponents in each term is n Coefficients are symmetric ( Pascal’sTriangle )

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Binomial Theorem

Binomial Theorem. Do you see anything?. 1 1 (n=0) a + b 1 1 (n=1) a 2 + 2ab + b 2 1 2 1 (n=2) a 3 + 3a 2 b + 3ab 2 + b 3 1 3 3 1 (n=3)

352 views • 13 slides

The Binomial Theorem

The Binomial Theorem. Unit 10.5. Binomial Theorem. (a + b) 0 = 1a 0 b 0 (a + b) 1 = 1a 1 b 0 + 1a 0 b 1 (a + b) 2 = 1a 2 b 0 + 2a 1 b 1 + 1a 0 b 2 (a + b) 3 = 1a 3 b 0 + 3a 2 b 1 + 3a 1 b 2 + 1a 0 b 3

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The Binomial Theorem

The Binomial Theorem. 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?. Notice that each entry in the triangle corresponds to a value n C r 0 C 0 1 C 0 1 C 1 2 C 0 2 C 1 2 C 2

335 views • 9 slides

The Binomial Theorem

The Binomial Theorem. Lecture 34 Section 6.7 Wed, Mar 28, 2007. The Binomial Theorem. Theorem: Given any numbers a and b and any nonnegative integer n ,. The Binomial Theorem. Proof: Use induction on n . Base case: Let n = 0. Then ( a + b ) 0 = 1 and

536 views • 22 slides

The Binomial Theorem

Combinatorics. The Binomial Theorem. 7.5. 1. Pascal’s Triangle and the Binomial Theorem. ( x + y ) 0. = 1. ( x + y ) 1. = 1 x + 1 y. = 1 x 2 + 2 xy + 1 y 2. ( x + y ) 2. = 1 x 3 + 3 x 2 y + 3 xy 2 + 1 y 3. ( x + y ) 3. ( x + y ) 4.

154 views • 7 slides

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