<s>[INST] Finding the remainder when a polynomial is divided by another polynomial. Find the remainder when $x^{100}$ is divided by $x^2 - 3x + 2$.
I tried solving it by first calculating the zeroes of $x^2 - 3x + 2$, which came out to be 1 and 2. 
So then, using the Remainder Theorem, I put both their values, and so the remainder came out to be $1 + 2^{100}$.
But the correct answer is $(2^{100} - 1)x + (2 - 2^{100})$. 
Can you please explain the exact process to reach the solution?
Thanks in advance. :) 
 [/INST] We write the Euclidean division:
$$x^{100}=(x^2-3x+2)Q(x)+ax+b$$
and notice that $1$ and $2$ are roots of $x^2-3x+1$ so


*

*let $x=1$ we get $1=a+b$

*let $x=2$ we get $2^{100}=2a+b$
so we find $a=2^{100}-1$ and $b=2-2^{100}$.

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