Question

Скоротіть дріб:
х¹¹+х¹⁰+х⁹+х⁸+х⁷+х⁶+х⁵+х⁴+х³+х²+х+1
----------------------------------------------------
(х³+1)(х⁶+1)

Answer 1

$\boxed {\ x^{n}-1=(x-1)(x^{n-1}+x^{n-2}+x^{n-3}+...+x^2+x+1)\ }\\\\\\x^{12}-1=(x-1)(x^{11}+x^{10}+x^9+...+x^2+x+1)\ \ \ \Rightarrow \\\\x^{11}+x^{10}+x^9+...+x^2+x+1=\dfrac{x^{12}-1}{x-1}\\\\\\\\\dfrac{x^{11}+x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1}{(x^3+1)(x^6+1)}=\dfrac{(x^{12}-1)}{(x-1)(x^3+1)(x^6+1)}=\\\\\\=\dfrac{(x^6-1)(x^6+1)}{(x-1)(x^3+1)(x^6+1)}=\dfrac{x^6-1}{(x-1)(x^3+1)}=\dfrac{(x^3-1)(x^3+1)}{(x-1)(x^3+1)}=\dfrac{x^3-1}{x-1}=\\\\\\=\dfrac{(x-1)(x^2+x+1)}{x-1}=x^2+x+1$