Long Division of Polynomials

Polynomial division is an essential algebraic process to simplify expressions and solve equations. Just as numerical division separates a number into quotient and remainder, polynomial long division partitions a polynomial into simpler components; in this context, the dividend is the polynomial being divided, the divisor is the expression dividing it, and the result is expressed in terms of a quotient and a remainder.

The division begins by arranging the dividend and divisor in standard form—terms in descending powers of the variable. The leading term of the dividend is divided by the divisor's leading term to determine the quotient's first term. This term is multiplied by the entire divisor, and the result is subtracted from the dividend. The next term in the dividend is brought down, and the process repeats until the remainder has a degree less than the divisor. The result is then expressed as:

For example, dividing:

the first term of the quotient is 6x, followed by −2, yielding the final quotient 6x - 2 and a remainder of 4. The result is written as:

This process can be generalized: for polynomials P(x) and D(x), there exist unique polynomials Q(x) and R(x) such that

where the degree of R(x) is less than that of D(x).