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 by a linear polynomial is equal to .

Examples

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Example 1

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Show that the remainder of the division of a polynomial   by a linear polynomial   is equal to  . Solution : Divide   by   like the following one.

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

As,  , thus the remainder is equal to  .

Example 2

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Show that the remainder of the division of a polynomial   by a linear polynomial   is equal to  . Solution : Divide   by   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,  , thus the remainder   is equal to  .

Proof

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Proposition

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If   is a polynomial of a positive degree and   is any definite number, the remainder of the division of   by   will be  

Proof

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The remainder of the division of a polynomial of a positive degree   by   is either 0 or a non-zero constant. Let the remainder is   and the quotient is  . Then, for every value of  ,

 · 

Putting   in the equation  , we get   ·  . Thus, the remainder of  ÷  is equal to  .