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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

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  1. The Binomial Theorem Unit 10.5

  2. Binomial Theorem (a + b)0 = 1a0b0 (a + b)1 = 1a1b0 + 1a0b1 (a + b)2 = 1a2b0 + 2a1b1 + 1a0b2 (a + b)3= 1a3b0 + 3a2b1 + 3a1b2 + 1a0b3 (a + b)4 = 1a4b0 + 4a3b1 +6a2b2 + 4a1b3 + 1a0b4 (a + b)5 =1a5b0 + 5a4b1 +10a3b2 +10a2b3 + 5a1b4 + 1a0b5 Complete the next 5 binomial expansions

  3. Practical Implication • Computer programming 1 0 • Physics: Comparing earths’ radius to its height or weight • Easier way to add numbers

  4. Practice Problems Page 629 1b (2x + 3y)5 Identify the polynomial that will be computed 1a5b0 + 5a4b1 +10a3b2 +10a2b3 + 5a1b4 + 1a0b5 32x5 + 240x4y +720x3y2+1080x2y3 + 810xy4 + 243y5

  5. Practice Problems Page 629 Guided 2a page 629 (2x – 7)3 Pascal 1a3b0 + 3a2b1 + 3a1b2 + 1a0b3 8x3 - 84x2 + 294x3 – 343 Page 633 Problems 1 - 10

  6. Find binomial coefficients Formula nCr = n! (n – r)!r! r = term in the coefficient r = k – 1 (x + y)9, 6th term n = 9 r = 5 nCr = 9! = 9! = 9*8*7*6*5*4*3*2*1 = 126 (9 – 5)!5! 4!5! 4*3*2*1*5*4*3*2*1

  7. Binomials with coefficients other than 1 (2x – 3y)8, x3y5 n = 8 r = 5 a = 2x b = (-3y) nCr = n! 8!→8*7*6 = 56 (n – r)!r! 3!5! 3*2*1 56(2x)3(-3y)5 = 56(8)(-243) = -108,864

  8. Exercises Page 633 Problems 11 - 18

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