Proof of the Polynomial Remainder Theorem

Synopsis: Written below is a brief description of the polynomial remainder theorem. The theorem has a wide range of applications spanning from Algebra to Number Theory. This depicts how important the polynomial remainder theorem truly is, and why it must be taught in all courses and is a great tool.

The remainder theorem states when a polynomial denoted as $f(x)$ is divided by $x-a$ for some value of $x$, whether real or unreal, the remainder of $\frac{f(x)}{x-a}=f(a)$ Written below is the proof of the polynomial remainder theorem.

All polynomials can be written in the form $f(x)=d(x)\cdot{q(x)}+r(x)$, where $d(x)$ is the divisor of the function/polynomial $f(x)$, $q(x)$ is the quotient. amd $r(x)$ is the remainder. Because the $deg$ $r=0$ or the $deg$ $r<deg$ $d$ and the fact that degrees must be whole numbers($0$ and the positive numbers), the $deg$ $r=0$, and so to speak, $r(x)$ is a constant, which we will denote as $b$.


Knowing this, we can write

$f(x)=d(x)\cdot{q(x)}+b$

$f(x)=(x-a)\cdot{q(x)}+b$

$f(a)=(a-a)\cdot{q(a)}+b$

$f(a)=b$

We have hereby proven when the quantity $x-a$ is divided into a polynomial $f(x)$ of any degree, the value of $f(a)=b$, where b is the remainder. The remainder must be a constant because $deg$ $r=0$.

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