Find value using the roots of polynomial

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Moon
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Find value using the roots of polynomial

Unread post by Moon » Wed Dec 08, 2010 3:24 am

It is most probably Canada 1995. Quite nice, and medium difficulty problem. Try it!

If $\alpha,\beta,\gamma$ are roots of $x^3-x-1$, then find
\[\frac{\alpha-1}{\alpha+1}+\frac{\beta-1}{\beta+1}+\frac{\gamma-1}{\gamma+1}\]
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

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Zzzz
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Re: Find value using the roots of polynomial

Unread post by Zzzz » Wed Dec 08, 2010 10:04 am

Just use $\sum\alpha=0, \sum\alpha\beta=-1, \alpha\beta\gamma=1$ ;)
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Moon
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Re: Find value using the roots of polynomial

Unread post by Moon » Wed Dec 08, 2010 11:32 am

Remember, one of the main purpose of the forum is to practice writing formal solutions. :)
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

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Zzzz
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Re: Find value using the roots of polynomial

Unread post by Zzzz » Wed Dec 08, 2010 12:25 pm

Ow... sorry. Actually its painful :P I will write formal solutions from now :)
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Re: Find value using the roots of polynomial

Unread post by Moon » Wed Dec 08, 2010 5:04 pm

Post more problems, if you have any. :)
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zadid xcalibured
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Re: Find value using the roots of polynomial

Unread post by zadid xcalibured » Thu Feb 28, 2013 2:24 am

Let $S=\frac{\alpha-1}{\alpha+1}+\frac{\beta-1}{\beta+1}+\frac{\gamma-1}{\gamma+1}$
$S+3=2 (\frac{\alpha}{\alpha+1}+\frac{\beta}{\beta+1}+\frac{\gamma}{\gamma+1})$
$S+3=\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}=2$
I posted in the 1st topic of algebra forum. :mrgreen:

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