Monic polynomial

In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the coefficient of the nonzero term of highest degree) is equal to 1. That is to say, a monic polynomial is one that can be written as

x

n

+

c

n

−

1

x

n

−

1

+

⋯

+

c

2

x

2

+

c

1

x

+

c

0

,

{\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{2}x^{2}+c_{1}x+c_{0},}

with

n

≥

0.

{\displaystyle n\geq 0.}

Uses

Monic polynomials are widely used in algebra and number theory, since they produce many simplifications and they avoid divisions and denominators. Here are some examples.

Every polynomial is associated to a unique monic polynomial. In particular, the unique factorization property of polynomials can be stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials.

Vieta's formulas are simpler in the case of monic polynomials: The kth elementary symmetric function of the roots of a monic polynomial of degree n equals

(

−

1

)

k

c

n

−

k

,

{\displaystyle (-1)^{k}c_{n-k},}

where

c

n

−

k

{\displaystyle c_{n-k}}

is the coefficient of the (n−k)th power of the indeterminate.

Euclidean division of a polynomial by a monic polynomial does not introduce divisions of coefficients. Therefore, it is defined for polynomials with coefficients in a commutative ring.

Algebraic integers are defined as the roots of monic polynomials with integer coefficients.