<s>[INST] Find the remainder when $x^{10}+1$ is divided by $(x^2+1)(x^2+x+1)$ Find the remainder when $x^{10}+1$ is divided by $(x^2+1)(x^2+x+1)$
I have done it until the the divisor is of second degree. 
But here the degree of the remainder is $4$
This means the remainder will be of the form.     : $ax^3+bx^2+cx+d$ which makes it difficult to solve. Is it possible to solve $4$ variants with $4$ equations?
 [/INST] Using Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd,
$$x^{10}+1\equiv0\pmod{x^2+1}$$
Again $x^3-1=(x-1)(x^2+x+1)$
$$x^{10}+1= x(x^9-1)+x+1\equiv x+1\pmod{x^2+x+1}$$
Now apply CRT
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