Polynomials - Trying to understand reasoning

For discussing Olympiad Level Algebra (and Inequality) problems
tn8964
Posts:1
Joined:Sat Feb 18, 2017 9:09 am
Polynomials - Trying to understand reasoning

Unread post by tn8964 » Sat Feb 18, 2017 9:49 am

Hi all,

My name is Tom and I'm a math enthusiast, solving Math Olympiad and Putnam problems. I'm trying to understand this reasoning. Feel free to correct it but put in a language in such a way that a child or a novice will understand it. Here's a problem:

For which n is the polynomial $x^n + x - 1$ divisible by:
(a) $x^2 -x +1$
(b) $x^3 -x +1$

I reasoned that of the polynomial of $x^n + x - 1$ is divisible by either a or b, then a and b zeros are also the zeros of the polynomial $x^n + x - 1$. I started with (a) and used the quadratic formula to get this:

$\epsilon=\frac{1\pm i\sqrt3}2$

I plug that into the polynomial $x^n +x -1$ and get this:

$\epsilon=1-\epsilon$

My problem is the solution said this:
$\epsilon_i^n=1-\epsilon_i=\epsilon_i^{-1}$

I don't get how subtracting 1-epsilon will get me $1/\epsilon$. Someone elaborate with simple analogy. Thank you
Last edited by SANZEED on Sat Feb 25, 2017 12:47 am, edited 1 time in total.
Reason: Please use $L^{A}T_{E}X$ while posting a topic/reply.

Post Reply