 Polynomial remainder theorem

Dividing a polynomial $f$ by $X-a$ leaves the constant polynomial remainder $f(a)$. Polynomial division makes the polynomial remainder constant because its polynomial degree must be below one; polynomial evaluation at $a$ identifies that constant. In particular, it is zero exactly when $a$ is a root of a polynomial of $f$, which is the factor theorem.^[ref]

In the worked example, division by $X-1$ leaves $6$ and division by $X+2$ leaves $12$. Division by their product leaves $-2X+8$, a polynomial remainder that stores both values at once. The argument also holds over a commutative ring for $a$ in that ring, because $X-a$ is a monic polynomial.

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 \( c \).