##  [Remainder Theorem](/remainder-theorem-0) 

 Definition

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

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

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

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

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

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

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

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

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.