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.
Properties Every nonzero univariate polynomial (polynomial with a single indeterminate) can be written
c n x n + c n − 1 x n − 1 + ⋯ c 1 x + c 0 , {\displaystyle c_{n}x^{n}+c_{n-1}x^{n-1}+\cdots c_{1}x+c_{0},}
where c n , … , c 0 {\displaystyle c_{n},\ldots ,c_{0}} are the coefficients of the polynomial, and the leading coefficient c n {\displaystyle c_{n}} is nonzero. By definition, such a polynomial is monic if c n = 1. {\displaystyle c_{n}=1.}
The product of monic polynomials is monic. The product of polynomials is monic if and only if the product of the leading coefficients of the factors equals 1. This implies that the monic polynomials in a univariate polynomial ring over a commutative ring form a monoid under polynomial multiplication. Two monic polynomials are associated if and only if they are equal, since the multiplication of a polynomial by a nonzero constant produces a polynomial with this constant as its leading coefficient. Divisibility induces a partial order on monic polynomials. This results almost immediately from the preceding properties.
Polynomial equations Let P ( x ) = 0 {\displaystyle P(x)=0} be a polynomial equation, where P is a univariate polynomial of degree n. If one divides all coefficients of P by its leading coefficient c n , {\displaystyle c_{n},} one obtains a new polynomial equation that has the same solutions. For example, the equation
2 x 2 + 3 x + 1 = 0 {\displaystyle 2x^{2}+3x+1=0}
is equivalent to the monic equation
x 2 + 3 2 x + 1 2 = 0. {\displaystyle x^{2}+{\frac {3}{2}}x+{\frac {1}{2}}=0.}
When the coefficients are unspecified, or belong to a field where division does not result in fractions (such as R , C , {\displaystyle \mathbb {R} ,\mathbb {C} ,} or a finite field), this reduction to monic equations may provide simplification. On the other hand, as shown by the previous example, when the coefficients are integers, the associated monic polynomial is generally more complicated. Therefore, primitive polynomials are often used instead of monic polynomials when dealing with integer coefficients.
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