Polynomial Division with three terms

Question

Find all numbers $n\geq1$ for which the polynomial $x^{n+1}+x^n+1$ is divisible by

(a) $x^2+x+1$

(b) $x^2-x+1$.

I tried to do long division, but I didn't really seem to be going anywhere. Any suggestions? thanks

Please include what you've tried in attempting to solve the problem. Perhaps playing around with long division might help? It's still doable with three terms. https://en.wikipedia.org/wiki/Polynomial_long_division — Shuri2060, Jul 06 '17 at 23:51

score 0 · Answer 1 · answered Jul 07 '17 at 00:12

0

Hint $ $ mod $\,x^2+x+1\!:\,\ x^3\equiv 1\,\Rightarrow\, x^n\equiv x^{n\bmod 3}$ and similarly for the other (negate $x)$

answered Jul 07 '17 at 00:12

Bill Dubuque

272,048

See here for some numerical examples. – Bill Dubuque Jul 07 '17 at 00:13

score 0 · Answer 2 · answered Jul 07 '17 at 00:13

Hint:

It is divisible by $x^2+x+1$ if and only the (complex) roots of $x^2+x+1$ are roots of $x^{n+1}+x^n+1$. Now the complex roots of the quadratic polynomial are the non-real cube roots of unity, $j$ and $j^2$. Calculate the successive powers of $j$ and plug them in $x^{n+1}+x^n+1$ to check.

Similarly, the roots of $x^2-x+1$ are the cube roots of $-1$, namely $-j$ and $-j^2$.

Polynomial Division with three terms

2 Answers2