maths notes (binomial expansion)

Binomial Expansion

Binomial expansion for positive integral indices [ (a+b)n ]

1. Equation for binomial expansion if n∈Z+¿¿

a.

(a+b)n=an+ n1 !an−1b+

n(n−1)2 !

an−2b2+…+n (n−1 ) (n−2 )… (n−r+1 )

r !an−rbr+…+

n (n−1 ) (n−2 )…2(n−1 )!

abn−1+bn

b. OR (a+b)n=(n0)an+(n1)an−1b+(n2)an−2b2+…+(nr )an−rbr+…+( nn−1)abn−1+(nn)bni. Where (nr )=n (n−1 ) (n−2 )… (n−r+1 )

r ! And r !=r (r−1 ) (r−2 )… (3)(2)(1)

2. Properties of a binomial expansion if n∈Z+¿¿

a. The expansion is a finite series with (n+1) termsb. The sum of powers of a and b in each term is always nc. The expansion is valid for all values of a and b

d. The general term is T r+1=(nr )an−rbr3. Tips for solving questions involving binomial expansion with positive integral indices

a. If asked to find the term with xa, consider the general equation, and find the term [

(r+1)th term] that has xa

b. There is a term with xa provided r∈N (r is a natural (non-negative, whole number)).

Else the coefficient of xa=0

Binomial expansion for non-positive/ non-integral indices [ (a+bx)m ]

1. Equation for binomial expansion if n∉Z+¿¿

a.

(a+bx )m=am(1+ bxa )m

=1+ n1!bx+

n (n−1 )2 !

(bx )2+ n(n−1 ) (n−2 )

3 !(bx )3+…+

n (n−1 ) (n−2 )… (n−r+1 )r !

(bx )r+…

2. Properties of a binomial expansion if n∉Z+¿¿

a. The expansion is called the Binomial Series and it’s an infinite series

b. To expand such a series, it must be in the form of (1+ bxa )m

. I.e. It needs to start with 1.

c. The term with xr=T r+1=n (n−1 ) (n−2 )… (n−r+1 )

r !(bx )r

i. (nr ) cannot be used when n∉Z+¿¿

d. The expansion is only valid if |bx|<13. Useful shortcut expansions

Done by Nickolas Teo Jia Ming, CG 12/11

a. (1+bx )−1=1−bx+(bx )2−(bx )3+…+(−1 )r(bx )r+…b. (1−x )−1=1+bx+ (bx )2+ (bx )3+…+(bx )r+…

Done by Nickolas Teo Jia Ming, CG 12/11