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
Binomial – two terms
(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 nCr
so we can see that tn,r = nCr = n!/(r!(n-r)!)
by Pascal’s formula we can see that
nCr = n-1Cr-1 + n-1Cr
Rewrite the following using Pascal’s Formula
10C4 18C8 + 18C9
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
Notice that there is one more term than the exponent number!
(a + b)n = nC0an + nC1an-1b+ nC2an-2b2 + … + nCran-rbr + … + nCnbn
(a + b)5
Try it with (3x – 2y)4
Factoring using the binomial theorem
Rewrite 1 + 10x2 + 40x4 + 80x6 + 80 x8 + 32x10 in the form (a + b)n
We know that there are 6 terms so the exponent must be five
The first term is 1
Therefore, a =
The final term is 32x10
Therefore, b =
Pg 293 # 1ace, 3ab, 4bc, 5ac, 8, 9ace,11ad,12a, 16a, 21