Nonmonic

A nonmonic polynomial is a polynomial whose leading coefficient is not equal to 1. This contrasts with a monic polynomial, whose leading coefficient is 1. The leading coefficient is the coefficient of the term with the highest degree that appears with a nonzero coefficient. In many contexts, especially over fields, the leading coefficient is a unit, so a polynomial is often considered up to multiplication by a nonzero constant; in such cases every nonzero polynomial is associated with a unique monic polynomial.

Over a field, any nonzero polynomial p(x) can be written as p(x) = a · q(x), where a is the leading coefficient of p and q(x) is monic. The monic polynomial q(x) has the same degree and the same zeros (in algebraic closures) as p(x), since p(x) and q(x) differ only by multiplication by a nonzero constant. Conversely, p(x) is nonmonic precisely when its leading coefficient a ≠ 1.

In rings where the leading coefficient is not a unit, dividing by the leading coefficient to obtain a monic polynomial may not be possible. When the leading coefficient is a unit, one can form the monic associate p*(x) = a^{-1} p(x), which is monic and preserves the roots and degree.

Nonmonic polynomials appear in various algebraic procedures, but many algorithms prefer monic representatives for canonical forms and simpler arithmetic. If a polynomial is scaled by a nonzero constant, its nonzero roots and factor structure over a field remain unchanged; only the leading coefficient and scale factor change.