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I have to find the remainder of the division of $X^n$ by $X^2-6X-16$ for all positive integer $n$. I know how to do a euclidian division with polynomials but I am confused with the $n$ since I don't know if $n>2$ or not. How do I do such a division?

moray95
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    Closely related to http://math.stackexchange.com/questions/602623/how-do-i-divide-a-polynomial-of-a-very-high-degree-by-a-polynomial-of-degree-2. – lhf Dec 29 '13 at 11:48

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Hint

Write $$X^n=(X^2-6X-16)Q(X)+aX+b\quad(*)$$ Replace $X$ in $(*)$ by the roots $-2$ and $8$ of $X^2-6X-16$ and solve the two equations of the unknowns $a$ and $b$.

Edit We find $$-2a+b=(-2)^n\quad;\quad 8a+b=8^n$$ hence $$a=\frac{8^n-(-2)^n}{10}\quad;\quad b=\frac{8^n+4(-2)^n}{5}$$

  • Using you method, I found $a=(2^n+8^n)/10$ and $b=(8^n-4.2^n)/5$. Is that right? – moray95 Dec 29 '13 at 12:14
  • @moray95 I edited my answer. –  Dec 29 '13 at 14:02
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    +1. Sami, regarding to your new avatar, I thin everything can be solved easily, even mathematical ones, by saying that holly words. و قال ارکبوا فیها بسم الله مجراها و مرساها – Mikasa Dec 29 '13 at 17:37