Question

Expand the following:

`(3x - 1/x)^3`

Answer 1

Given: `(3x - 1/x)^3`

Step-wise calculation using the binomial expansion formula for (a – b)³ = a³ – 3a²b + 3ab² – b³:

1. Here, `a = 3x` and `b = 1/x`.

2. Compute each term:

a³ = (3x)³ = 27x³ 

`3a^2b = 3 xx (3x)^2 xx 1/x`

`3a^2b = 3 xx 9x^2 xx 1/x`

3a²b = 27x

`3ab^2 = 3 xx 3x xx (1/x)^2`

`3ab^2 = 3 xx 3x xx 1/x^2`

`3ab^2 = 9/x`

`b^3 = (1/x)^3`

`b^3 = 1/x^3`

3. Substitute these into the expansion formula with correct signs (note the minus signs alternate):

`(3x - 1/x)^3 = a^3 - 3a^2b + 3ab^2 - b^3`

`(3x - 1/x)^3 = 27x^3 - 27x + 9/x - 1/x^3`