cubic polynomial

[stuart clark]

If \(\displaystyle \alpha,\beta\) and \(\displaystyle \gamma\) be the roots of the equation \(\displaystyle x^3+x-1 = 0\). Then Calculate value of \(\displaystyle \frac{\beta}{\alpha}+\frac{\gamma}{\beta}+\frac{\alpha}{\gamma}\)

 

[tkhunny]

What have you tried? You didn't find the roots, did you? That would be quite painful, I would think, since only one is Real.

Write the entire expression over the common denominator \(\displaystyle \alpha\beta\gamma\)

Expand this and contemplate the coefficients: \(\displaystyle (x-\alpha)(x-\beta)(x-\gamma)\)

The first thing you should notice is that \(\displaystyle \alpha\beta\gamma\;=\;-1\)

Let's see what you get.