The Student Room Group
Uni students - Complete our survey >>
Uni students - Complete our survey >>
Uni students - Complete our survey >>

(C4) Finding coefficients from binomial expansion

Question (the supplied answers show that a=25, b=63):

Assuming that the expansion of the following is valid, find the values of a and b:

( (1+x)^3 / (1-x)^4 ) = 1 + 7x + ax^2 + bx^3


---

I calculated (1+x)^3 to expand as 1+3x+3x^2 + x^3

and (1-x)^-4 to expand as 1/(1- 4x +6x^2 - 4x^3 + x^4).

This would give me


( 1+ 3x+ 3x^2 + x^3 ) / (1- 4x +6x^2 - 4x^3 + x^4)

but I don't know where to go from there to solve for a and b (coefficients of x^2 and x^3).
Reply 1
Avatar for B_9710
B_9710
18
Original post by SkyJP
Question (the supplied answers show that a=25, b=63):

Assuming that the expansion of the following is valid, find the values of a and b:

( (1+x)^3 / (1-x)^4 ) = 1 + 7x + ax^2 + bx^3


---

I calculated (1+x)^3 to expand as 1+3x+3x^2 + x^3

and (1-x)^-4 to expand as 1/(1- 4x +6x^2 - 4x^3 + x^4).

This would give me


( 1+ 3x+ 3x^2 + x^3 ) / (1- 4x +6x^2 - 4x^3 + x^4)

but I don't know where to go from there to solve for a and b (coefficients of x^2 and x^3).


Do the expansion of (1+x)^3 and (1-x)^(-4) and then multiply them. This will give you the expansion where you can find a and b. You shouldn't be dividing the two products you should be multiplying.
Reply 2
Avatar for B_9710
B_9710
18
What I mean by this is don't think of (1-x)^4 as 1/(1-x)^4. Expand it using the binomial theorem as (1-x)^-4
Reply 3
Avatar for Subspace
Subspace
OP
8
Original post by B_9710
What I mean by this is don't think of (1-x)^4 as 1/(1-x)^4. Expand it using the binomial theorem as (1-x)^-4



I see, so, essentially, (1+x)^3 / (1-x)^4 becomes (1+x)^3 (1-x)^-4 , which then becomes polynomial multiplication after I've expanded them both?
Reply 4
Avatar for B_9710
B_9710
18
Original post by SkyJP
I see, so, essentially, (1+x)^3 / (1-x)^4 becomes (1+x)^3 (1-x)^-4 , which then becomes polynomial multiplication after I've expanded them both?


Exactly.

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you