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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. ( 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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The Binomial Theorem

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  1. Chapter 7 Combinatorics 7.5 The Binomial Theorem 7.5.1 MATHPOWERTM 12, WESTERN EDITION

  2. 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

  3. The Binomial Theorem The Binomial Theorem is a formula used for expanding powers of binomials. Each term of the answer is the product of three first-degree factors. For each term of the answer, an a and/or b is taken from each first-degree factor. (a + b)3 = (a + b)(a + b)(a + b) = • The first term has no b. It is like choosing no b from three b’s. • The combination • The second term has one b. It is like choosing one b from three b’s. • The combination • The third term has two b’s. It is like choosing two b’s from three • b’s. The combination. • The fourth term has three b’s. It is like choosing three b’s from • three b’s. The combination (a + b)3 7.5.3

  4. Pascal’s Triangle and the Binomial Theorem The numerical coefficients in a binomial expansion can be found in Pascal’s triangle. Pascal’s Triangle Using Combinatorics Pascal’s Triangle 1st Row 1 2nd Row 1 1 2 1 3rd Row 1 3 3 1 4th Row 1 4 1 6 1 4 5th Row 1 5 1 5 10 10 6th Row 7.5.4

  5. 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= 3C0a3 - 0b0 + 3C1a3 - 1b1+ 3C2a3 - 2b2 + 3C3a3 - 3b3 7.5.5

  6. Binomial Expansion - Practice Expand the following. a) (3x + 2)4 b) (2x- 3y)4 7.5.6

  7. Binomial Expansion - Practice c) 7.5.7

  8. Finding a Particular Term in a Binomial Expansion a) Find the eighth term in the expansion of (3x - 2)11. tk + 1 = nCkan - kbk b) Find the middle term of (a2 - 3b3)8. tk + 1 = nCkan - kbk 7.5.8

  9. Finding a Particular Term in a Binomial Expansion Find the constant term of the expansion of tk + 1 = nCkan - kbk 7.5.9

  10. Finding a Particular Term in a Binomial Expansion Find the numerical coefficient of the x11 term of the expansion of 7.5.10

  11. Assignment Suggested Questions Pages 356 and 357 4, 12, 14, 20-30, 33, 34, 36, 37, 40 abd 7.5.12

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