## Find value using the roots of polynomial

For discussing Olympiad Level Algebra (and Inequality) problems
Moon
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Joined: Tue Nov 02, 2010 7:52 pm
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### Find value using the roots of polynomial

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

learn how to write equations, and don't forget to read Forum Guide and Rules.

Zzzz
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Joined: Tue Dec 07, 2010 6:28 am
Location: 22° 48' 0" N / 89° 33' 0" E

### Re: Find value using the roots of polynomial

Just use $\sum\alpha=0, \sum\alpha\beta=-1, \alpha\beta\gamma=1$ Every logical solution to a problem has its own beauty.

Moon
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Joined: Tue Nov 02, 2010 7:52 pm
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### Re: Find value using the roots of polynomial

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

learn how to write equations, and don't forget to read Forum Guide and Rules.

Zzzz
Posts: 172
Joined: Tue Dec 07, 2010 6:28 am
Location: 22° 48' 0" N / 89° 33' 0" E

### Re: Find value using the roots of polynomial

Ow... sorry. Actually its painful I will write formal solutions from now Every logical solution to a problem has its own beauty.

Moon
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Contact:

### Re: Find value using the roots of polynomial

Post more problems, if you have any. "Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

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. 