In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
For any polynomials f(x) and g(x), with g(x) not identical to zero, there exist unique polynomials q(x) and r(x) such that
with r(x) having smaller degree than g(x).
Polynomial long division finds the quotient q(x) and remainder r(x) given a numerator f(x) and nonzero denominator g(x). The problem is written down like a regular (nonalgebraic) long division problem:
All terms with exponents less than the largest one must be written out explicitly, even if their coefficients are zero.
Find
The problem is written like this:
The quotient and remainder can then be determined as follows:
The polynomial above the bar is the quotient, and the number left over (−123) is the remainder.
The long division algorithm learned in elementary arithmetic classes can be viewed as a special case of the above algorithm.
