Euclidean division of polynomials

The Euclidean Division Theorem

For any polynomials A(x) and B(x), where B is non-zero, there exists a unique pair (Q(x), R(x)) such that: A(x) = B(x) * Q(x) + R(x), where deg(R) < deg(B) (or R = 0) Q is the quotient, R the remainder. The principle is the same as for Euclidean division of integers.

Practical method (step-by-step division)

Example: divide A(x) = 2x³ - 3x² + 4x - 1 by B(x) = x - 2. Step 1: 2x³ / x = 2x², so we multiply B by 2x² and subtract this from A. Step 2: We repeat the operation on the partial remainder until we obtain a degree strictly less than deg(B). Result: Q(x) = 2x² + x + 6 and R = 11, so A(x) = (x - 2)(2x² + x + 6) + 11.

Special case: division by (x - a)

When B(x) = x - a, the remainder R is a constant equal to A(a) (remainder theorem). Horner’s scheme then allows Q and R to be calculated very quickly; this method is widely used in numerical computation and computer science.

Common pitfall

Do not confuse Euclidean division (which may have a remainder) with exact division (where the remainder is zero, i.e. when B divides A). Always check that deg(R) < deg(B): a remainder of the same or higher degree than B indicates a calculation error in the subtraction steps.