Question

If x^2-3x+1=0 then find the value of x^3+1/x^3

Answer 1

We have to know that,

$x+\dfrac {1}{x}=k\quad\implies\quad x^3+\dfrac {1}{x^3}=k^3-3k$

So,

$x^2-3x+1=0\\\\x^2+1=3x\\\\\dfrac {x^2+1}{x}=3\\\\x+\dfrac {1}{x}=3\implies\\\\x^3+\dfrac {1}{x^3}=3^3-3\cdot 3\\\\x^3+\dfrac {1}{x^3}=\mathbf {18}$