All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Unknown Polynomials (Posted on 2005-12-23)
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution | Comment 1 of 5
`            nLet 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
Please log in:
 Login: Password: Remember me: Sign up! | Forgot password

 Search: Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information