Using the roots of polynomial finding the value of sum.

Question

If $a,b$ and $c$ are the roots of $x^{3}+px^{2}+qx+r$, then how can we find the value of $\displaystyle \sum \frac{b^{2}+c^{2}}{bc}$.

Answer 1

I think you are asking for $$\frac{a^2+b^2}{ab}+\frac{b^2+c^2}{bc}+\frac{c^2+a^2}{ca},\tag{$1$}$$ or perhaps twice this quantity. If you bring Expression $(1)$ to a common denominator, you will get $$\frac{a^2c+b^2c+b^2 a+c^2 a+c^2b+a^2b}{abc}.$$

Note that $$(a+b+c)(ab+bc+ca)=a^2c+b^2c+b^2 a+c^2 a+c^2b+a^2b+3abc.$$

Thus $$\frac{a^2+b^2}{ab}+\frac{b^2+c^2}{bc}+\frac{c^2+a^2}{ca}=\frac{(a+b+c)(ab+bc+ca)-3abc}{abc}.$$ Everything term on the right-hand side is expressible simply in terms of the coefficients.