The Polynomial Remainder Theorem is a fundamental result in algebra that relates to the division of polynomials. It states that if a polynomial \( f(x) \) is divided by a linear polynomial of the form \( (x - c) \), the remainder of this division is equal to the value of the polynomial evaluated at \( c \).
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Dividing a polynomial by leaves the constant polynomial remainder . Polynomial division makes the polynomial remainder constant because its polynomial degree must be below one; polynomial evaluation at identifies that constant. In particular, it is zero exactly when is a root of a polynomial of , which is the factor theorem.[ref]
Fields and Galois Theory by James S. Milne. Fields and Galois Theory, version 5.10, Chapter 1, p. 10, §1.7: division by X−a and evaluation of the constant remainder
In the worked example, division by leaves and division by leaves . Division by their product leaves , a polynomial remainder that stores both values at once. The argument also holds over a commutative ring for in that ring, because is a monic polynomial.