Prove that the polynomial is $g(x,y)(x^2 + y^2 -1)^2 + c$











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This is from a Brazilian math contest for college students (OBMU):



Let $f(x,y)$ be a polynomial in two real variables such that the polynomials



$$frac{partial f}{partial x}(x,y)$$



$$frac{partial f}{partial y}(x,y)$$



are divisible by $x^2+y^2-1$. Prove that there's a polynomial $g(x,y)$ and a constant $c$ such that



$$f(x,y) = g(x,y)(x^2+y^2 -1)^2 +c$$






real-analysis multivariable-calculus contest-math multivariate-polynomial







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    up vote
    0
    down vote

    favorite












    This is from a Brazilian math contest for college students (OBMU):



    Let $f(x,y)$ be a polynomial in two real variables such that the polynomials



    $$frac{partial f}{partial x}(x,y)$$



    $$frac{partial f}{partial y}(x,y)$$



    are divisible by $x^2+y^2-1$. Prove that there's a polynomial $g(x,y)$ and a constant $c$ such that



    $$f(x,y) = g(x,y)(x^2+y^2 -1)^2 +c$$






    real-analysis multivariable-calculus contest-math multivariate-polynomial







    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      This is from a Brazilian math contest for college students (OBMU):



      Let $f(x,y)$ be a polynomial in two real variables such that the polynomials



      $$frac{partial f}{partial x}(x,y)$$



      $$frac{partial f}{partial y}(x,y)$$



      are divisible by $x^2+y^2-1$. Prove that there's a polynomial $g(x,y)$ and a constant $c$ such that



      $$f(x,y) = g(x,y)(x^2+y^2 -1)^2 +c$$






      real-analysis multivariable-calculus contest-math multivariate-polynomial







      share|cite|improve this question















      This is from a Brazilian math contest for college students (OBMU):



      Let $f(x,y)$ be a polynomial in two real variables such that the polynomials



      $$frac{partial f}{partial x}(x,y)$$



      $$frac{partial f}{partial y}(x,y)$$



      are divisible by $x^2+y^2-1$. Prove that there's a polynomial $g(x,y)$ and a constant $c$ such that



      $$f(x,y) = g(x,y)(x^2+y^2 -1)^2 +c$$






      real-analysis multivariable-calculus contest-math multivariate-polynomial




      real-analysis multivariable-calculus contest-math multivariate-polynomial






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 2 days ago









      Jean Marie

      28.1k41848




      28.1k41848










      asked 2 days ago









      Rafael Deiga

      657310




      657310



























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