Question

How do you use the Binomial Theorem to expand `(1/X + X)^6`?

Answer 1

`(1/x+x)^6=1/x^6+6/x^4+15/x^2+20+15x^2+6x^4+1/x^6`

Explanation:

Binomial expansion of `(x+a)^6` is

`x^6+^6C_1x^5a+^6C_2x^4a^2+^6C_3x^3a^3+^6C_4x^2a^4+^6C_5xa^5+^6C_6a^6`

And hence binomial expansion of `(1/x+x)^6` or `(x+1/x)^6` is

`x^6+^6C_1x^5(1/x)+^6C_2x^4(1/x)^2+^6C_3x^3(1/x)^3+^6C_4x^2(1/x)^4+^6C_5x(1/x)^5+^6C_6(1/x)^6`

or `x^6+6/1xx x^4+(6xx5)/(1xx2)xx x^2+(6xx5xx4)/(1xx2xx3)+(6xx5xx4xx3)/(1xx2xx3xx4)xx 1/x^4+(6xx5xx4xx3xx2)/(1xx2xx3xx4xx5)xx 1/x^6`

or `x^6+6x^4+15x^2+20+15/x^2+6/x^4+1/x^6`

or `1/x^6+6/x^4+15/x^2+20+15x^2+6x^4+1/x^6`