Polynomials - Trying to understand reasoning
Posted: 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
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