Question

If (x + 1/x)^2 = 3 what is the value of x^3 + 1/x^3

Answer 1

(x + 1/x)² = 3                 | √

x + 1/x = √3

According to short multiplication formula:

(a + b)³ =a³ + 3a²b + 3ab² + b³

x³ + 1/x³ = (x)³ + (1/x)³ = (x + 1/x)³ - 3x²/x - 3x/x² =

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

= (√3)³ - 3√3 = 3√3 - 3√3 = 0

If (x + 1/x)² = 3 the value of x³ + 1/x³ is 0.