Polynomial Long Division

Polynomial division is an algebraic operation that divides a polynomial $ A(x) $ (called the dividend) by another non-zero polynomial $ B(x) $ (called the divisor), in order to obtain a quotient $ Q(x) $ and a remainder $ R(x) $ such that $$ A(x) = B(x) \times Q(x) + R(x) $$

This operation generalizes Euclidean division of integers to polynomials.

The long division procedure involves three steps:

— a) Divide the highest-degree term of the dividend by the highest-degree term of the divisor; the resulting partial quotient becomes the first term of Q(x).

— b) Multiply the entire divisor by this partial quotient

— c) Subtract this product from the dividend, which gives a partial remainder

Repeat steps a,b,c until the degree of the remainder is less than that of the divisor (or until the remainder is zero)

If $ B(x) $ divides $ A(x) $ without remainder, then $ B(x) $ is a factor of $ A(x) $

If the divisor is a monomial or a simple monomial of the form $ ax^k $, it is possible to divide each term of the dividend by that monomial separately.

This is a simpler division than long division, which is often used as a special case.