A polynomial division problem can be checked by verifying that the dividend equals the product of the quotient and the divisor, plus the remainder. When dividing a polynomial function $f(x)$ by a binomial divisor of the form $x - c$, this relationship can be written in function notation as $f(x) = q(x)(x - c) + r$, where $q(x)$ is the quotient and $r$ is the remainder. Evaluating this function at $x = c$ involves substituting $c$ for $x$, resulting in the equation $f(c) = q(c)(c - c) + r$. Because the term $(c - c)$ simplifies to $0$, the equation becomes $f(c) = q(c)(0) + r$, which reduces to $f(c) = r$. This mathematical proof demonstrates that evaluating the original polynomial function at $c$ yields the exact value of the remainder.