# On dividing a polynomial p(x) by a non-zero polynomial q(x),

Question:

On dividing a polynomial p(x) by a non-zero polynomial q(x), let g(x) be the quotient and r(x) be the remainder, than p(x) = q(x)⋅g(x) + r(x), where

(a) r(x) = 0 always
(b) deg r (x) g(x) always
(c) either r(x) = 0 or deg r(x) g(x)
(d) r(x) = g(x)

Solution:

(c) either $r(x)=0$ or $\operatorname{deg} r(x)<\operatorname{deg} g(x)$

By division algorithm on polynomials, either $r(x)=0$ or $\operatorname{deg} r(x)<\operatorname{deg} g(x)$.