My Math Forum A problem of Cyclic polynomials

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 June 2nd, 2010, 07:07 PM #1 Newbie   Joined: Jun 2010 Posts: 1 Thanks: 0 A problem of Cyclic polynomials Yesterday my little sis of level 9 came to me with a problem of her textbook. When I saw the problem I told her, "I'm busy now, come later". Then I tried this problem overnight and failed to solve. Would anybody help me to save my goodwill in math? Please. If $\frac{x^{2}-yz}{a}= \frac{y^{2}-zx}{b} = \frac{z^{2}-xy}{c}\neq 0$ then prove that $\left(x+y+z\right)\left(a+b+c\right)= ax+by+cz$
 June 4th, 2010, 04:57 AM #2 Senior Member   Joined: Sep 2008 Posts: 150 Thanks: 5 Re: A problem of Cyclic polynomials Hi Boka, that is just an exercise in patient calculations: First define$\gamma:=\frac {a}{x^2-yz}=\frac{b}{y^2-xz}=\frac{c}{z^2-xy}$, which is possible, as the fractions in the assumption are not zero. This gives you $a=\gamma\cdot (x^2-yz)$ and likewise for b and c. You have to show that (x+y+z)(a+b+c)-(ax+by+cz) vanishes so simply plug in the expressions for a,b,c and see that the summands cancel each other. (Of course it might be convenient, to cancel what is possible before plugging in the expressions for a,b,c.) Best regards Peter

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