1

Let $R(x)$ be a remainder upon dividing $x^{44}+x^{33}+x^{22}+x^{11} +1$ by the polynomial $x^4 +x^3 +x^2 +x +1$. Find: $R(1)+2R(2)+3R(3)$. Answer provided is $0$

mnulb
  • 3,701

5 Answers5

11

Hint \begin{align} & {{x}^{55}}-1=({{x}^{11}}-1)({{x}^{44}}+{{x}^{33}}+{{x}^{22}}+{{x}^{11}}+1) \\ & {{x}^{5}}-1=(x-1)({{x}^{4}}+{{x}^{3}}+{{x}^{2}}+{{x}^{1}}+1) \\ \end{align} $$\frac{{{x}^{44}}+{{x}^{33}}+{{x}^{22}}+{{x}^{11}}+1}{{{x}^{4}}+{{x}^{3}}+{{x}^{2}}+{{x}^{1}}+1}=\frac{{{x}^{55}}-1}{{{x}^{5}}-1}\times \frac{(x-1)}{({{x}^{11}}-1)}$$

Behrouz Maleki
  • 11,342
5

$$x^{44}+x^{33}+x^{22}+x^{11}+1=\frac{x^{55}-1}{x^{11}-1}=\frac{(x^5-1)\sum_{k=0}^{10}x^{5k}}{x^{11}-1}\\=\frac{x^5-1}{x-1}\cdot\frac{\sum_{k=0}^{10}x^{5k}}{\sum_{k=0}^{10}x^k}=(x^4+x^3+x^2+x+1)\cdot\color{blue}{\frac{\sum_{k=0}^{10}x^{5k}}{\sum_{k=0}^{10}x^k}}$$

The blue fraction is actually a polynomial because the only eleventh root of $1$ which is also a fifth root of $1$ is $1$ itself. Or, if you prefer, because $\operatorname{gcd}(x^{11}-1,x^5-1)$ is obviously $x-1$.

4

The thing that makes this easy is $x^4 + x^3 + x^2 + x + 1$ divides $x^5 - 1$. Thus, if $$\begin{align} f(x) &\equiv g(x) & &\pmod{x^5 - 1} \\ g(x) &\equiv h(x) & &\pmod{x^4 + x^3 + x^2 + x + 1} \end{align}$$ then $$ f(x) \equiv h(x) \pmod{x^4 + x^3 + x^2 + x + 1} $$ and it's easy to find a low degree $g(x)$ to use as the intermediate, since $x^5 \equiv 1 \pmod{x^5-1}$.

But even without that, you can use the usual tricks for modular arithmetic; e.g. some of the content of How do I compute $a^b\,\bmod c$ by hand? applies here too.

Community
  • 1
3

Hint $\,\ (x\!-\!1)f\, =\, x^{\large 5}-1\,\ $ for $\,\ f =\, x^{\large 4}+x^{\large 3}+\cdots+1,\,\ $ therefore

$\qquad\quad \begin{eqnarray}{\rm mod}\ f\!:\,\ x^{\large \color{#c00}5}\equiv 1\ \Rightarrow\ &&x^{\large 4+\color{#c00}5I}+\!&&x^{\large 3+\color{#c00}5J}+\!&&x^{\large 2+\color{#c00}5K}+\!&&x^{\large 1+\color{#c00}5L}+\!&&x^{\large\color{#c00} 5M}\\ \equiv\,\ &&x^{\large 4}\quad + &&x^{\large 3}\quad + &&x^{\large 2} \quad + &&x\quad\ + &&1\,\equiv\, f\,\equiv\, 0\end{eqnarray}$

Bill Dubuque
  • 282,220
  • Above we used five times that $\ x^{\large A + \color{#c00}5B} = x^{\large A} (x^{\large \color{#c00}5})^{\large B}\equiv x^{\large A}(1^{\large B})\equiv x^{\large A}.\ $ Note that this method requires no knowledge of any special factorization formulas (except $,(x!-!1)f = x^5-1),,$ so it will work much more generally. – Bill Dubuque Jul 11 '16 at 22:42
3

Long division of $x^{44}+x^{33}+x^{22}+x^{11}+1$ by $x^4+x^3+x^2+x+1$ gives a quotient of $$ x^{40}-x^{39}+x^{35}-x^{34}+x^{30}-x^{28}+x^{25}-x^{23}+x^{20}-x^{17}+x^{15}-x^{12}+x^{10}-x^{6}+x^{5}-x+1$$ and no remainder. It is not a lot of work: only five lines.

lhf
  • 221,500