What Are the Possible Values of r in This Viete Relations Problem?

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SUMMARY

The discussion focuses on determining the possible values of r in the context of Viete relations, specifically for the equation involving nonzero real numbers x, y, and z. The equation is reformulated to find r such that x²y + y²z + z²x = rxyz. The participants clarify the correct formulation of the equations and explore the implications of Viete's relations, leading to the conclusion that the relationships among the roots can provide conditions on r. The final solution hinges on the correct interpretation of the polynomial formed by the roots x, y, and z.

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  • Understanding of Viete's formulas and their application in polynomial equations.
  • Familiarity with polynomial functions and their properties.
  • Knowledge of real number properties and algebraic manipulation.
  • Ability to work with equations involving multiple variables and their relationships.
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  • Study Viete's relations in depth to understand their implications in polynomial equations.
  • Explore the properties of symmetric polynomials and their applications in algebra.
  • Learn about the derivation and manipulation of cubic polynomials.
  • Investigate conditions for the existence of real roots in polynomial equations.
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Mathematics students, algebra enthusiasts, and anyone interested in solving polynomial equations using Viete's relations.

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[SOLVED] Viete relations problem

Homework Statement


Find all real numbers r for which there is at least one triple (x,y,z) of nonzero real numbers such that

x^2 y + yz^2 + z^2 x = xy^2 + yz^2 + zx^2 = rxyz

Homework Equations


http://en.wikipedia.org/wiki/Viète's_formulas

The Attempt at a Solution


This is equivalent to finding the possible values of r+s+t = 1/r + 1/s + 1/t where r,s,t are real but I don't see how that leads to a solution.Fix r and assume that x,y,z exist. Let f(t) = t^3 + at^2 + bt+c be the monic polynomial with
x,y,z as its zeros. By assumption c is not zero. Its not hard to show that ab = (3+2r)c and a^3 = x^3+ y^3+z^3 + (3+2r)c using Viete's relations. But I am not sure what to do with those or how to get any sort of condition on r.

Please just provide a hint.
 
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Just to be sure, did you type out the equations correctly? Or is it supposed to be \sum x^2 y = rxyz instead?
 
morphism said:
Just to be sure, did you type out the equations correctly? Or is it supposed to be \sum x^2 y = rxyz instead?

I did mess up. Change yz^2 to y^2 z on the LHS. Anyway I already peeked at the solution.
 

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