I have got stuck in it. although the problem is not so difficult but I am willing to see from the expertise if there is any innovative way to handle the problem.
Suppose that $p(x),d(x)$ be polynomials with integer coeffcients with degrees $n, d$ respectively where $d\leq n$. By Euclidian algorithm we can obtain the quotient and remainder say $Q(x), R(x)$.
My query is: will it be possible to find the sum of $Q(x)$ and $R(x)$ at $x=1$ without obtaining explicitly $Q(x), R(x)$ through division algorithm ?
My so far experience is not so innovative. What I have found is some thing else: $Q(1)+R(1)\neq$ Quo$\left(\frac{p(1)}{d(1)}\right)+$Rem$\left(\frac{p(1)}{d(1)}\right)$ where Quo$(\frac{p(x)}{d(x)})$ and Rem$(\frac{p(x)}{d(x)})$ are the quotient and the remiander upon the divison of $p(x)$ by $d(x)$ respectively.
So which means I am still forced to find first $Q(x)$ and $R(x)$ explicitly and then evaluation at $x=1$.
But is it not possible to get the same result without finding $Q(x), R(x)$ ?
Please help me.
