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?

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Presentation transcript:

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 3 C o 3 C 1 3 C 2 3 C 3 so we can see that t n,r = n C r = n!/(r!(n-r)!)

by Pascal’s formula we can see that n C r = n-1 C r-1 + n-1 C r Rewrite the following using Pascal’s Formula 10 C 4 18 C C 9

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 = 8a a 2 b +54ab b 3 Notice that there is one more term than the exponent number! From Dan

(a + b) n = n C 0 a n + n C 1 a n-1 b + n C 2 a n-2 b 2 + … + n C r a n-r b r + … + n C n b n or Expand (a + b) 5 Try it with (3x – 2y) 4

Factoring using the binomial theorem Rewrite x x x x x 10 in the form (a + b) n We know that there are 6 terms so the exponent must be five Step 1

The final term is 32x 10 Step 2 Therefore, b = The first term is 1 Therefore, a = Step 3

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