## The remainder after division

Show that if a polynomial $$p(x)$$ is divided by the linear polynomial $$x-c$$, then the remainder is equal to $$p(c)$$.
The division rule is $$p(x)=q(x)(x-c)+r(x)$$.
The degree of $$r$$ is smaller than the degree of the linear polynomial $$(x-c)$$, i.e. it is a constant factor $$r(x)=r$$.
If we substitute $$x=c$$ into $$p(x)=q(x)(x-c)+r$$ we get $$p(c)=q(c)(c-c)+r=r$$, as it was required.