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Unknown Polynomials (Posted on 2005-12-23) Difficulty: 5 of 5
Determine all polynomials L(x) satisfying the undernoted equation:
(x-243) L(3x) = 243 (x-1) L(x) for all x.

See The Solution Submitted by K Sengupta    
Rating: 2.6667 (3 votes)

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Solution | Comment 1 of 5

            n
Let L(x) = SUM [a_i*x^i]. Then
           i=0
  (x - 243)*L(3x) = 243*(x - 1)*L(x)  becomes
               n                         n
  3^5*(x - 1)*SUM [a_i*x^i] = (x - 3^5)*SUM [3^i*a_i*x^i]
              i=0                       i=0
                            or
   n                             n
  SUM [3^5*a_i*(3^i - 1)*x^i] = SUM [a_i*(3^i - 3^5)*x^(i+1)]
  i=0                           i=0
                            or
   n                            n+1
  SUM [3^5*a_i*(3^i - 1)*x^i] = SUM [a_(i-1)*(3^(i-1) - 3^5)*x^i]
  i=1                           i=1
                            or
   n                             n
  SUM [3^5*a_i*(3^i - 1)*x^i] = SUM [a_(i-1)*(3^(i-1) - 3^5)*x^i]
  i=1                           i=1
                                + a_n*(3^n - 3^5)*x^(n+1)
Let n = 5 and we get the difference equation for i = 1 to 5:
         3^(i-1) - 3^5
  a_i = ---------------
         3^5*(3^i - 1)
Using a computer for the dirty work we get the following monic polynomial: 
  L(x) = x^5 - 363x^4 + 32,670x^3 - 882,090x^2 + 7,144,929x - 14,348,907
 

  Posted by Bractals on 2005-12-23 18:35:52
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