Proof of an Integer polynomial property Could any one give a basic algebra proof that for any polynomial P with integer coefficients, P(a)−P(b) is divisible by a−b. 
 A: It is enough to show that the statement is true for monomials.
$$a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\cdots + ab^{n-2}+ b^{n-1})$$
A: Let $Q(x) = P(a)-P(b+x)$ then $a-b$ is a root of $Q$ so $x - (a-b)|Q(x)$ and setting $x=0$ gives $a-b|P(a)-P(b)$.
A: Hint $\rm\,\ mod\ a\!-\!b\!:\,\ a\equiv b\:\Rightarrow\: P(a)\equiv P(b),\ $ for all $\rm\:P\in \Bbb Z[x],\:$ by congruence sum, product rules.
Remark $\ $ For completeness here is a proof of the implication by induction on the degree of $\rm\,P.\:$ Clear if $\rm\:deg\ P = 0.\:$ Else $\rm\:P(x) = c + x\,Q(x)\:$ for $\rm\:c = P(0),\ Q(x)\in\Bbb Z[x]\:$ with $\rm\:deg\ Q < deg\ P.\:$ Therefore, by induction, $\rm\ a\equiv b\:$ $\Rightarrow$ $\rm\:Q(a)\equiv Q(b)\:$ $\Rightarrow$ $\rm\:P(a) = c + a\, Q(a)\equiv c + b\, Q(b)\equiv P(b),\ $ since, by the product and sum rules for congruences we have
$$\begin{eqnarray}\rm a&\equiv&\rm b,\ \ \ Q(a)&\equiv&\rm Q(b)\,\ &\Rightarrow&\rm\,\ a\, *\, Q(a)&\equiv&\rm b \,*\, Q(b)\\ 
\rm c &\equiv&\rm c,\ \ aQ(a)&\equiv&\rm bQ(b)\,\ &\Rightarrow&\rm\,\ c+aQ(a)&\equiv&\rm c+bQ(b) \end{eqnarray}$$
Notice that the effect of the induction is to lift  congruence-preservation from the basic addition and multiplication operations to compositions of such, i.e. polynomial expressions. The innate structure will become clearer when you learn about polynomial rings and quotient (residue) rings.
