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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

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The Binomial Theorem

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  1. The Binomial Theorem

  2. 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?

  3. Notice that each entry in the triangle corresponds to a value nCr 0C0 1C0 1C1 2C02C12C2 3Co3C13C23C3 so we can see that tn,r = nCr = n!/(r!(n-r)!)

  4. by Pascal’s formula we can see that nCr = n-1Cr-1 + n-1Cr Rewrite the following using Pascal’s Formula 10C4 18C8 + 18C9

  5. The coefficients of each term in the expansion of (a + b)n correspond to the terms in row n of Pascal’s Triangle. Therefore you can write these coefficients in combinatorial form. Lets look at (2a + 3b)3 = 8a3 + 36a2b +54ab2 + 27b3 From Dan Notice that there is one more term than the exponent number!

  6. (a + b)n = nC0an + nC1an-1b+ nC2an-2b2 + … + nCran-rbr + … + nCnbn or Expand (a + b)5 Try it with (3x – 2y)4

  7. Factoring using the binomial theorem Rewrite 1 + 10x2 + 40x4 + 80x6 + 80 x8 + 32x10 in the form (a + b)n Step 1 We know that there are 6 terms so the exponent must be five

  8. Step 2 The first term is 1 Therefore, a = Step 3 The final term is 32x10 Therefore, b =

  9. Homework Pg 293 # 1ace, 3ab, 4bc, 5ac, 8, 9ace,11ad,12a, 16a, 21

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