# monic polynomial

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Related to monic polynomial: monic equation, Irreducible polynomial

## monic polynomial

[¦mō·nik ‚päl·ə′nō·mē·əl]
(mathematics)
A polynomial in which the coefficient of the term of highest degree is +1 and the coefficients of the other terms are integers.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Since [c.sub.1](z), [c.sub.2](z), ..., [c.sub.m](z) are arbitrary monic polynomials subject, however, to conditions (6) and (7), we assume without any loss of generality that
(1) We are dealing only with monic polynomials, that is, with those for which the coefficient of [x.sup.2] is 1.
By using (1.1) we obtain a system of |n| linear equations for the |n| unknown coefficients of the monic polynomial [P.sub.n](x) = [[summation].sup.|n|.sub.k=0] [a.sub.k],nxk, [a.sub.|n|,n] = 1.
Our goal here is the first interpretation for the coefficients of [[PHI].sub.n](x), which applies more generally to the coefficients of any monic polynomial f(x) in Z[x].
Such a monic polynomial may not exist or may not be unique.
A sequence of monic polynomials [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is said to be a Laguerre-Hahn sequence of class s if it is orthogonal with respect to a linear functional w satisfying
(To justify this claim, the main things to note are that (a) a monic polynomial that is irreducible over Z is also irreducible over [Z.sub.2] = [F.sub.2], and (b) if P' is P with coefficients reduced mod 2, and H(P',Q') [less than or equal to] d, then by adding even integers to the coefficients of Q' we can arrive at a polynomial Q such that |P - Q| [less than or equal to] d.)
Since V(0) [not equal to] 0, and since the roots of a monic polynomial identify it uniquely, the factorization in (1) is unique.
This implies that we can as well consider the monic polynomial
Let [q.sub.n] (x) be the classical monic polynomial, orthogonal with respect to a classical weight function w (x) on (a, b).

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