# Proof of the Polynomial Remainder Theorem

The remainder theorem states when a polynomial denoted as is divided by for some value of , whether real or unreal, the remainder of Written below is the proof of the polynomial remainder theorem.

All polynomials can be written in the form $f(x)=d(x)\cdot\q(x)+r(x)$ (Error compiling LaTeX. ! Undefined control sequence.), where is the divisor of the function/polynomial , is the quotient. amd is the remainder. Because the or $deg r\less\deg d$ (Error compiling LaTeX. ! Undefined control sequence.) and the , degrees must be whole numbers, and so . So to speak, is a constant. We denote this constant .

Knowing this, we can write

$f(x)=d(x)\cdot\q(x)+b$ (Error compiling LaTeX. ! Undefined control sequence.)

$f(x)=(x-a)\cdot\q(x)+b$ (Error compiling LaTeX. ! Undefined control sequence.)

$f(a)=(a-a)\cdot\q(a)+b$ (Error compiling LaTeX. ! Undefined control sequence.)

We have hereby proven when the quantity is divided into a polynomial of any degree, the value of , where b is the remainder. The remainder must be a constant because .