What Is a Monic Polynomial?

A monic polynomial is a mathematical expression that consists of coefficients and a single variable, with the leading coefficient equal to one. The leading coefficient is found in the term that contains the variable with the highest degree or exponent.

Monic polynomials are characterized by having only a single variable, hence the term "univariate polynomial". The default orientation of a monic polynomial involves the terms arranged from highest to lowest degree. The term that contains the variable with the highest exponent is written first, then the next term with the second highest exponent until the term with the lowest degree. In other words, the usual arrangement of a monic polynomial is in descending powers.

Any polynomial can be converted to a monic polynomial by dividing all the terms by the coefficient of the highest order term. In this case, there is a possibility that the coefficients of the lower order terms will appear as fractions. This does not matter for monic polynomials, as long as the leading coefficient has a value of one.

Some references use the term "monic" even in polynomials that contain several variables, although it is fairly uncommon. In these variants, the "last" variable may be used as a reference for the order of the terms. For instance, the polynomial xy^2 - 3x^3y + 3 is a monic polynomial in reference to the variable y.