Monic polynomial
In algebra, a monic polynomial is a univariate polynomial in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form
If a polynomial has only one indeterminate (univariate polynomial), then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A univariate polynomial in x of degree n then takes the general form displayed above, where
are constants, the coefficients of the polynomial.
Here the term cnxn is called the leading term, and its coefficient cn the leading coefficient; if the leading coefficient is 1, the univariate polynomial is called monic.
The set of all monic polynomials (over a given (unitary) ring A and for a given variable x) is closed under multiplication, since the product of the leading terms of two monic polynomials is the leading term of their product. Thus, the monic polynomials form a multiplicative semigroup of the polynomial ring A[x]. Actually, since the constant polynomial 1 is monic, this semigroup is even a monoid.
The restriction of the divisibility relation to the set of all monic polynomials (over the given ring) is a partial order, and thus makes this set to a poset. The reason is that if p(x) divides q(x) and q(x) divides p(x) for two monic polynomials p and q, then p and q must be equal. The corresponding property is not true for polynomials in general, if the ring contains invertible elements other than 1.
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