The Remainder Theorem states that if a polynomial function $f(x)$ is divided by a binomial of the form $x - c$, then the remainder of that division is exactly equal to the value of the function evaluated at $c$, which is $f(c)$. This theorem can be logically derived by expressing the division in function notation: to get the dividend $f(x)$, multiply the quotient $q(x)$ by the divisor $x - c$, and add the remainder $r$. This gives the equation $f(x) = q(x)(x - c) + r$. If this equation is evaluated at $c$, it yields $f(c) = q(c)(c - c) + r$, which simplifies to $f(c) = q(c)(0) + r$, leaving $f(c) = r$.