The Binomial Expansion

Edexcel IAL WMA12/01/P2/Oct 2019/Q3 (Binomial Expansion)

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

1+x412

Giving each coefficient in its simplest form.

(3)

• Find the term independent of x in the expansion of

x2+8x5 1+x412

(3)

SOLUTION​​ 

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

an=120112x40=111=1

n1an-1 b=121111x41=121 x4=3x

n2an-2 b2=122110x42=661x216=338 x2

n3an-3 b3=12319x43=220 1x364 =5516 x3

1+x412=1+3x+38x2+5516x3+…

b-​​ 

x2+8x5 1+x412=x2x5+8x51+3x+38x2+5516x3+…

=1x3+8x51+3x+38x2+5516x3+…

The term independent of x is the one which is constant. So we need to find the constant terms in the expansion. One of the constant term we will get when​​ 1x3​​ is multiplied with​​ 5516x3. And second, when​​ 8x5​​ is multiplied by term in​​ x5​​ of​​ 1+x412. ​​ 

When​​ 1x3​​ is multiplied with​​ 5516x3

=1x3 x5516 x3=5516

When​​ 8x5​​ is multiplied by term in​​ x5​​ of​​ 1+x412. ​​ 

=8x5 x125 17 x45

=8x5 x 7 92 x 1 xx51024

=9916

Hence, the term independent of x is

5516+9916=778