If X^2-5x+1=0, the find the value of x^3+1/x^3?

Question

If X^2-5x+1=0, the find the value of x^3+1/x^3?

Answer 1

x^2 + 1 = 5x

x + 1/x = 5

also squaring we have .. . .

(x + 1/x)^2 = 25

x^2 + 2 + 1/x^2 = 25

now

x^3 + 1/x^3 = (x + 1/x) (x^2 - 1 + 1/x^2)

thus

x^3 + 1/x^3 = (5) (23 - 1) = 110

.. . .. .