The Binomial Theorem
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Transcript of 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 nCr
0C0
1C0 1C1
2C0 2C1 2C2
3Co 3C1 3C2 3C3
so we can see that tn,r = nCr = n!/(r!(n-r)!)
by Pascal’s formula we can see thatnCr = n-1Cr-1 + n-1Cr
Rewrite the following using Pascal’s Formula10C4 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!
From Dan
(a + b)n = nC0an + nC1an-1b + nC2an-2b2 + … + nCran-rbr + … + nCnbn
or
n
r
rrnrn
n baCba0
)(
Expand (a + b)5
Try it with (3x – 2y)4
Factoring using the binomial theoremRewrite 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
Step 1
The final term is 32x10
Step 2
Therefore, b =
The first term is 1
Therefore, a =
Step 3
HomeworkPg 293 # 1ace, 3ab, 4bc, 5ac, 8, 9ace,11ad,12a, 16a, 21