# how to find the remainder when a polynomial $p(x)$ is divided my another polynomial $q(x)$

i was solving the question from the book IIT FOUNDATION AND OLYMPIAD - X and i was solving the problems of polynomials-III. so on the page number 136, there is a question (question 17) given below:

The remainder when $x$^100 is divided by $x^2-3x+2$ is:

a) $(2$^100$-1)x + (-2$^100$+2)$

b) $(2$^100$+1)x + (-2$^100$-2)$

c) $(2$^100$-1)x + (-2$^100$-2)$

d) none

as far as i tried to find the remainder, i tried long division method but it was getting more and more complicated, then i used systematic method of division but i can't get the corret option what is the correct option. please explain me how did you find the remainder. thanks

and yes its answer is option (a)

Hint Write $x^{100}= (x^2-3x+2)q(x) + ax+b$. Now plug $x=1$ and $x=2$ to find $a$ and $b$.
• @anni, the remainder is $ax+b$. – lhf Jun 21 '14 at 15:31
Hint $\,\ f\equiv g\ \ ({\rm mod} (x\!-\!1)(x\!-\!2))\ \iff x\!-\!1,x\!-\!2\mid f\!-\!g\ \iff f(1) = g(1),\ f(2) = g(2)$
• @anni By above, with $\,f(x) = x^{100},\,$ an answer $\,g(x)\,$ is correct $\iff g(1) = 1,\ g(2) = 2^{100}.\ \$ – Bill Dubuque Jun 21 '14 at 15:34