The Binomial Expansion

Edexcel IAL WMA12/01/P2/June/Oct 2020/Q1 (Binomial Expansion)

• Find the first 4 terms, in ascending powers of x, of the binomial expansion of​​ 

2-x410

giving each term in its simplest form.​​ 

(4)​​ 

• Hence find the constant term in the series expansion of​​ 

3-1x22-x410

(3)

SOLUTION​​ 

a-​​ Using binomial theorem to find the first 4 terms of the expression.​​ 

an=100210-x40=1 ×1024 ×1=1024

n1an-1 b=10129-x41=10 ×512 ×-x4=-1280x

n2an-2b2=10228-x42=45 ×256×x216=720x2

n3an-3b3=10327-x43=120 ×128×-x364=-240x3

2-x410=1024-1280x+720x2-240x3+….

b-​​ 

3-1x22-x410=9-6x+1x21024-1280x+720x2-240x3+…

There are going to be three constant terms. One of the constant term we get when two constant​​ of the two bracket gets multiplied. Whereas the other is formed when the​​ 6x​​ is multiplied to the​​ 1280x​​ term. And, the third term when​​ 1x2​​ gets multiplied with​​ 720x2.​​ 

When two constant of the two bracket gets multiplied

=9×1024

When the​​ 6x​​ is multiplied to the​​ 1280x​​ term

=-6x×-1280x=6×1280

When​​ 1x2​​ gets​​ multiplied with​​ 720x2

=1x2×720x2=720

Hence, the constant term of the series expansion is​​ 

=9×1024+6×1280+720

=17616