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

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    I have a feeling this was supposed to be, find the value of $\frac {a^2+b^2+c^2} {abc}\, ,$ Is this correct?2012-06-06

1 Answers 1

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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.

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    That makes sense. +12012-06-06