Author |
Topic: A polynomial (Read 392 times) |
|
wonderful
Full Member
Posts: 203
|
 |
A polynomial
« on: Jul 2nd, 2008, 9:28pm » |
 Quote  Modify
|
Find a polynomial P(x) of degree n ( x in R) which has real root(s) and P(x).P(2x^2)=P(2x^3+x) for every x in R
Have A Great Day!
|
|
 IP Logged |
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender: 
Posts: 1948
|
|
Re: A polynomial
« Reply #1 on: Jul 2nd, 2008, 11:09pm » |
Quote Modify
|
Do you mean it has a real root, or that all roots are real?
If the latter, there is no such polynomial, because if x is a non-zero real root, then 2x3+x is another root of larger norm.
And if the former, there is also no such polynomial, because the only possible real root is x=0, and then we can show inductively that all coefficients must be 0.
|
| « Last Edit: Jul 2nd, 2008, 11:41pm by Eigenray » |
IP Logged |
|
|
|
wonderful
Full Member
Posts: 203
|
|
Re: A polynomial
« Reply #2 on: Jul 2nd, 2008, 11:40pm » |
Quote Modify
|
For the moment, we can assume that all roots of P(x) are real. I think you forgot the term P(x) in the product P(x).P(2x^2+2), didn't you?
Have A Great Day!
|
| « Last Edit: Jul 2nd, 2008, 11:47pm by wonderful » |
IP Logged |
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: A polynomial
« Reply #3 on: Jul 2nd, 2008, 11:58pm » |
Quote Modify
|
No, I think you forgot the term +2 in P(x).P(2x2+2). 
If x is a root of P, then 2x3+x is also a root. Since there are only finitely many roots, if x is real it must be 0.
So we can't have every root real.
|
| « Last Edit: Jul 2nd, 2008, 11:59pm by Eigenray » |
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print