Timeless Theorems of Mathematics/Polynomial Remainder Theorem

The Polynomial Remainder Theorem is an application of Euclidean division of polynomials. It is one of the most fundamental and popular theorems of Algebra. It states that the remainder of the division of a polynomial $P(x)$ by a linear polynomial $(x-r)$ is equal to $P(r)$.

Examples

Example 1

Show that the remainder of the division of a polynomial $f(x)=x^{2}-2x+2$ by a linear polynomial $(x-1)$ is equal to $f(1)$. Solution : Divide $f(x)=x^{2}-2x+2$ by $(x-1)$ like the following one.

x - 1 ) x^2 - 2x + 2 ( x - 1
        x^2 - x
        ------------
            - x + 2
            - x + 1
        ------------
                  1

As, $f(1)=1^{2}-2(1)+2=1$, thus the remainder is equal to $f(1)$.

Example 2

Show that the remainder of the division of a polynomial $F(x)=ax^{2}+bx+c$ by a linear polynomial $(x-m)$ is equal to $F(m)$. Solution : Divide $F(x)=ax^{2}+bx+c$ by $(x-m)$ like the following one.

x-m ) ax^2+bx+c ( ax+am+b
      ax^2-amx
      ------------------
           amx+bx+c
           amx     -am^2
      ------------------
               bx+c+am^2
               bx-bm
      ------------------
               am^2+bm+c

As, $f(m)=am^{2}+bm+c$, thus the remainder $am^{2}+bm+c$ is equal to $f(m)$.

Proof

Proposition

If $P(x)$ is a polynomial of a positive degree and $a$ is any definite number, the remainder of the division of $P(x)$ by $(x-a)$ will be $P(a)$

Proof

The remainder of the division of a polynomial of a positive degree $P(x)$ by $(x-a)$ is either 0 or a non-zero constant. Let the remainder is $R$ and the quotient is $Q(x)$. Then, for every value of $x$,

$P(x)=(x-a)$·$Q(x)+R....(i)$

Putting $x=a$ in the equation $(i)$, we get $P(a)=0$ · $Q(a)+R=R$. Thus, the remainder of $P(x)$÷$(x-a)$ is equal to $P(a)$.