Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
What is the sum over? – Thomas Jun 6 '12 at 15:55
I have a feeling this was supposed to be, find the value of $$\frac {a^2+b^2+c^2} {abc}\, ,$$ Is this correct? – process91 Jun 6 '12 at 16:10
up vote 7 down vote accepted

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.

share|cite|improve this answer
That makes sense. +1 – Thomas Jun 6 '12 at 16:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.