Find the value of $\frac{\alpha^{2014}+\beta^{2014}+\alpha^{2016}+\beta^{2016}}{\alpha^{2015}+\beta^{2015}}$ for the roots of $x^2-3x+1$

Question

I got this problem from my textbook, not school. I tried various methods but was unable to solve the problem.

Let $\alpha$ and $\beta$ be the roots of the polynomial $x^2-3x+1$. I need to find the value of $$ \frac{\alpha^{2014}+\beta^{2014}+\alpha^{2016}+\beta^{2016}}{\alpha^{2015}+\beta^{2015}}$$

Answer 1

The trick is to realize that $\alpha^2 + 1 = 3 \alpha$, and $\beta^2 + 1 = 3\beta$.

Using this, we have $\alpha^{2016} + \alpha^{2014}= \alpha^{2014}(3\alpha) = 3\alpha^{2015}$

Combining this with the analogous expression in $\beta$, we can write

$$\frac{\alpha^{2016} + \beta^{2016} + \alpha^{2014} + \beta^{2014}}{\alpha^{2015} + \beta^{2015}} = \frac{3(\alpha^{2015} +\beta^{2015})}{(\alpha^{2015} +\beta^{2015})} = 3$$