Well, we know from the remainder left by dividing by (x-a) = f(a). So you get a linear system.

Say our polynomial p(x) = . Putting in

dividing p(x) by (x-1), the remainder =1. so

like wise and the remainder left by divind x-6 = . Lets call the remainder p(x) divided by x-6 = mSo we get

We get

etc...

which is the linear system

. Solving this you get

since for a deg 5 polynomial, its coeff cant be 0, The solution is the remainder when p(x) is divided by x-6 is any number except 6.