The Binomial Expansion

Coach Name: Sir Muhammad Abdullah Shah

Edexcel IAL WMA12/01/P2/Jan 2021/Q4 (Binomial Expansion)

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

2+px6

where p is a constant. Give each term in​​ simplest form.​​ 

(4)

Given that in the expansion of​​ 

3-12x2+px6

the coefficient of​​ x2 is -34​​ 

  • find the possible values of p.​​ 

(4)

SOLUTION​​ 

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

Finding the first three terms separately and will then arrange​​ them in the ascending order of power of x.​​ 

an=6026px0=1 x 64 x 1=64

n1an-1 b=6125px1=6 x 32 x px=192 px

n2an-2b2=6224px2=15 x 16 x p2x2=240p2x2

2+px6=64+192px+240p2x2+ 

 

b- ​​ It is given that the coefficient of​​ x2​​ is​​ -34.​​ 

3-12x2+px6=3-12x64+192px+240p2x2+

=364+3192px+3240p2x2-12x64-12x192px-12x240p2x2+

Not expanding further as on multiplying we will get x with a power greater than 2. Which is not required rightnow.​​ 

It is given in the question that the​​ coefficient of​​ x2​​ is​​ -34.

Where, on expansion of​​ 3-12x2+px6, we get the coefficient of​​ x2​​ as​​ 3240p2x2​​ and​​ -12x192px.​​ 

coefficient of x2=3×240+-12192p

coefficient of x2=720 p2-96 p

 -34=720p2-96p

 -3=2880 p2-384 p

2880 p2-384p+3=0

p=--384±-3842-42880322880

p=18, p=1120