The rule that when a polynomial p(x) is divided by the linear polynomial (x−r), the remainder equals p(r); hence evaluation at r yields the remainder of that division.

Principle

Polynomial division by a linear divisor is captured by evaluation: the remainder of division by (x−r) is a constant equal to the value of the polynomial at r, so division and evaluation commute in this special case.

Demonstration

Example: p(x) = x^3 − 4x + 1 divided by x − 2 leaves remainder p(2) = 8 − 8 + 1 = 1. Synthetic division will produce the same constant remainder and simultaneously produce the quotient polynomial.

Misapplication

Assuming the same evaluation rule holds for divisors of higher degree; using p(r) as 'the remainder' when dividing by a non-linear polynomial is incorrect because remainders then are polynomials of positive degree.

Consequence

Provides a fast test for roots (remainder zero) and a practical computational shortcut: to check whether r is a root compute p(r) rather than perform full polynomial division, and enables synthetic division techniques.

Reversal

If the remainder of dividing p(x) by (x−r) is zero then p(r)=0, recovering the Factor Theorem. The contrast is that nonzero remainders quantify the failure of r to be a root.

Boundary

Applies to polynomial division over fields and integral domains; when coefficients lie in rings with zero divisors evaluation may not reflect division behavior. The theorem addresses division by linear polynomials only.

Semantic Tension

Overlap with the Factor Theorem creates tension: both relate roots and division by (x−r) but the Remainder Theorem emphasizes the numerical remainder p(r) while the Factor Theorem emphasizes factorization (remainder zero).

Synthesis

The Remainder Theorem ties evaluation to division: evaluating p at r yields the exact constant remainder of dividing by (x−r), giving an efficient criterion for root testing and a computational route to quotient extraction.