Division Algorithm

 

Let f(x) and  be two polynomial functions and suppose that g(x)  is a non-zero polynomial. Then there exists unique polynomial functions q(x)  and r(x) such that

  f(x) = g(x).q(x)+r(x)

Holds for all values of x ,r(x) and  is either zero constant or a polynomial of degree lower than the degree of g(x)

 

Example:

Divide 6x4+5x3+4x-4 by 2x2+x-1

Using long division, we have the following:

Since 4x-3 has a lower degree than 2x2+x-1, we can stop here and conclude that 3x2+x+1 is the quotient polynomial and 4x-3 is the remainder polynomial, implying

6x4+5x3+4x-4 = (2x2+x-1) (3x2+x+1)+ 4x-3 

It is also clear that in the division algorithm, f(x) is the dividend polynomial, g(x) is the divisor polynomial, q(x) is the quotient polynomial and r(x) is the remainder polynomial. Therefore, the identity f(x)=g(x).q(x)+r(x) expresses the fact that

(dividend) = (divisor)(quotient) + remainder