Editing Monic polynomial (section)

== Uses ==
{{hatnote|In this section, "polynomial" is used as a shorthand for [[univariate polynomial]], and, unless explicitly stated the coefficients of the polynomials belong to a fixed [[field (mathematics)|field]].}}

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 elements|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 polynomial]]s.''

[[Vieta's formulas]] are simpler in the case of monic polynomials: ''The {{mvar|i}}th [[elementary symmetric function]] of the [[polynomial root|roots]] of a monic polynomial of degree {{math|n}} equals <math>(-1)^ic_{n-i},</math> where <math>c_{n-i}</math> is the coefficient of the {{math|(n−i)}}th power of the [[indeterminate (variable)|indeterminate]].''

[[Euclidean division of polynomials|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 integer]]s are defined as the roots of monic polynomials with integer coefficients.