Find the remainder when ##(x-3)^200 - (x-2)^100## is divided by ##x^2 -5x+6##?

##r(x) = -2x+5##

Calling

##P(x) = (x-3)^200 - (x-2)^100## and ##p(x) =x^2 -5x+6 = (x-2)(x-3) ##

we have

##P(x) = p(x)Q(x)+r(x)## in which

##Q(x)## is the quocient polynomial and ##r(x) = a x + b## is the remainder

Then we have

##P(2) = 1 = 2 a + b## and ##P(3) = -1 = 3 a + b##

Solving for ##a,b## we have ##a = -2, b= 5## so

##r(x) = -2x+5##