Easy Approach to the Binomial Theorem

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  • Topic: Pascal's triangle, Binomial theorem, Coefficient
  • Pages : 3 (263 words )
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  • Published : November 27, 2012
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Binomial Theorem Formula Explained

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
……………………………………………………. They are identical to Pascal Δs 0C0
1C0 1C1
2C0 2C1 2C2
3C0 3C1 3C2 3C3
4C0 4C1 4C2 4C3 4C4
5C0 5C1 5C2 5C3 5C4 5C5
………………………………………………………………………………………….. Take the example of (a+b)5

(a+b)5 = 1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5

(a+b)5 = 5C0a5b0 + 5C1a4b1 + 5C2a3b2 + 5C3a2b3 + 5C4ab4 + 5C5b5 T1 T2 T3 T4 T5 T6
T0+1 T1+1 T2+1 T3+1 T4+1 T5+1

T4 = T3+1=5C3a2b3=5C3a5-3b3
n=5, r =3 for Tr+1= nCran-rbr
The general term in the binomial expansion
Tr+1= nCran-rbr
5C3a2b3
Why 5C3 ?
Note 5C3 = 5C2 =5C2 3C3 =10 ways:
(a+b) (a+b) (a+b) (a+b) (a+b)
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Exercise

Find the terms as indicated in the following binomial expansions: (1) (1+3x)7 the 5th term when it is expanded in ascending powers of x

(2) (x-2)8 the 3rd term hen it is expanded in descending powers of x

Expand the following in ascending powers of x as far as the 3rd term

(3) (1+2x)8

(4) [pic]

In the following, find the indicated term.
(5) [pic] constant term

(6) [pic] constant term

(7) [pic]...
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