Polynomial division and root multiplicity

To divide a polynomial f by a nonzero polynomial g over a field, subtract a multiple of g chosen to cancel the leading term of f. Repeat with the remaining polynomial. Each nonzero remainder has smaller degree than the previous one, so the process must terminate. The outcome is a quotient q and a remainder r whose degree is smaller than that of g, unless r is zero.

For uniqueness, suppose two such decompositions exist. Their difference says that g times the difference of quotients equals the difference of remainders. If the quotient difference were nonzero, the left side would have degree at least the degree of g, whereas the right side would have smaller degree. This is impossible in a polynomial ring over a field. Over a general coefficient ring, division still works for a monic divisor, or more generally one with a unit leading coefficient.

Divide by x minus a. The remainder must be constant, and evaluating at a shows that this constant equals f evaluated at a. Therefore a is a root exactly when x minus a divides f. If several distinct roots occur, remove their linear factors one at a time. A previously removed factor evaluates nonzero at a different root, so cancellation allows the argument to continue. A nonzero degree-d polynomial over a domain has at most d distinct roots.

The coefficient hypothesis matters. In the ring of residues modulo eight, the polynomial x squared minus one has four distinct roots: one, three, five, and seven. This does not violate the root bound because that coefficient ring has zero divisors. Also, over a finite field, distinct formal polynomials can define the same function on the field. The polynomial x to the p minus x vanishes at every element of the prime field without being the zero polynomial.

The multiplicity of a root is the largest power of its linear factor dividing the polynomial. A double real root may touch the horizontal axis without crossing, but this picture is not the definition and does not apply over a finite field. Differentiating a factored polynomial shows that every repeated root is also a root of its formal derivative. Conversely, a root shared by f and its derivative has multiplicity at least two.

Recovering the exact multiplicity from successive ordinary derivatives requires care in positive characteristic. Factorials can become zero in the field, so the familiar derivative test for the first nonzero derivative may fail. For example, in characteristic p the derivative of x to the p is zero although the polynomial is not constant. Factor divisibility remains the reliable definition in every characteristic; derivative-based shortcuts must state their assumptions.