Question

If x² − 4x + 1 = 0, find `x^3 + 1/x^3`.

Answer 1

x² − 4x + 1 = 0

⇒ x² + 1 = 4x

⇒ `(x^2 + 1)/x = 4`

⇒ `x + 1/x = 4`

Let’s take a = x, b = `1/x`,

Using the identity:

a³ + b³ = (a + b)³ − 3ab (a + b)

Thus,

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

`x^3 + 1/x^3 = (4)^3 - 3 (4)`

= 64 − 12

∴ `x^3 + 1/x^3 = 52`