We define the recursive polynomial $T_n(x)$ as follows: $T_0(x)=1$ $T_1(x)=x$ $T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$. a) find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$. b) find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$. Proposed by Morteza Saghafian
Problem
Source: Iran 3rd round 2011-algebra exam-p1
Tags: inequalities, algebra, polynomial, algebra proposed
Rust
07.09.2011 13:45
$T_n(x)=ch(narcch(x))$. b) If n even $T_n(x)\not >0$ for all x. If n is odd unique real root is $x=0$.
goodar2006
07.09.2011 14:00
the problem asks to find all the roots, not all the real roots.
anonymouslonely
14.09.2011 20:05
I obtained $ 2T_{n}=(x+\sqrt{x^{2}+1})^{n}+(x+\sqrt{x^{2}-1})^{n} $. how to go on?
goodar2006
14.09.2011 20:20
there is another way for thinking on this problem. just see the name of the topic.....
anonymouslonely
15.09.2011 05:56
maybe...but I am a beginner. can somebody post a solution?
goodar2006
15.09.2011 07:57
just notice that $T_n(x)=\frac{G_n(x)}{i^n}$ where $G_n(x)$ is the $n$th chebyshevs polynomial.
anonymouslonely
15.09.2011 15:28
what is the Cebisev polynomial?
goodar2006
15.09.2011 18:26
chebyshev polynomials
AnhIsGod
02.12.2012 05:50
I want a full solution!
Plops
15.05.2019 03:37
Is it just me or is this problem similar to CMIMC $2016$ Algebra Problem $10$? Also, does anyone have a solution. I think I might have.