Question

Find the product:

`(a - 1/a)(a^2 + 1/a^2 + 1)`

Answer 1

Given: `(a - 1/a)(a^2 + 1/a^2 + 1)`

Stepwise calculation:

1. Expand the product: 

`(a - 1/a)(a^2 + 1/a^2 + 1)`

= `a(a^2 + 1/a^2 + 1) - 1/a(a^2 + 1/a^2 + 1)`

2. Multiply terms inside:

= `(a^3 + a xx 1/a^2 + a) - (a^2/a + 1/a xx 1/a^2 + 1/a)`

3. Simplify powers:

= (a³ + a^–1 + a) – (a¹ + a^–3 + a^–1)

4. Group like terms:

= a³ + a^–1 + a – a – a^–3 – a^–1

5. Cancel terms (a) and (a^–1):

= a³ – a^–3

So, the product `(a - 1/a)(a^2 + 1/a^2 + 1) = a^3 - 1/a^3`.