Find the remainder when $$ x^{2008} + 2008x + 2008 $$ is divided by $$ x^2-3x+2.$$
I tried equating the divisor to 0, and used some polynomial reduction, but couldn't get to a proper solution. Here's what I did: $$ x^{2008} + 2008x + 2008 = (x-1)(x-2)Q(x) + ax+b $$ I substituted $ x=1, 2 $ And solved for a and b, I got $$a= 2^{2008}+2007, b = 2010-2^{2008}$$ What should I do next? Any help will be appreciated.
$$P(x)=x^{2008}+2008x+2008=Q(x)(x^2-3x+2)+ax+b =Q(x)(x-2)(x-1)+ax+b $$
$$P(1)=4017=a+b$$
$$P(2)=2^{2008}+6024=2a+b$$
$$a=2^{2008}+2007, b =2010-2^{2008}$$
$$R(x)=(2^{2008}+2007)x+2010-2^{2008}$$