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Polynomial division is similar to the division of integers we are familiar with. When we divide one number by another, we get a quotient and a remainder.
For example, when dividing by :
Here:
The same concept applies to polynomials.
The division algorithm states the relationship between the dividend polynomial, the divisor, the quotient, and the remainder.
If (the dividend) and (the divisor) are two polynomials, with , then there exist unique polynomials (the quotient) and (the remainder) such that:
or it can be written as:
with the condition that the degree of is less than the degree of , or (zero remainder).
Terminology:
Illustrative Example:
The division of by yields:
This can be written in two forms according to the algorithm:
Fractional form:
Multiplication form:
Note that the degree of the remainder (, degree ) is less than the degree of the divisor (, degree ).
We can prove the correctness of the second form above by multiplying the quotient by the divisor and then adding the remainder.
Prove that .
Let's expand the right side:
Since the right side equals the left side, the equation is proven true.
