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 edit
Example 1 edit
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 edit
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 edit
Proposition edit
If is a polynomial of a positive degree and is any definite number, the remainder of the division of by will be
Proof edit
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 .