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Let () be a polynomial equation, where P is a univariate polynomial of degree n. If one divides all coefficients of P by its leading coefficient, one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial. For example, the equation
Given a quadratic polynomial of the form + + it is possible to factor out the coefficient a, and then complete the square for the resulting monic polynomial. Example: + + = [+ +] = [(+) +] = (+) + = (+) + This process of factoring out the coefficient a can further be simplified by only factorising it out of the first 2 terms.
In linear algebra, the Frobenius companion matrix of the monic polynomial = + + + + is ... The explicit formula for the eigenvectors ...
Monic morphism, a special kind of morphism in category theory Monic polynomial , a polynomial whose leading coefficient is one A synonym for monogenic , which has multiple uses in mathematics
If () is a monic polynomial in one variable with coefficients in a unique factorization domain (or more generally a GCD domain), then a root of that is in the field of fractions of is in . [ note 5 ] If R = Z {\displaystyle R=\mathbb {Z} } , then it says a rational root of a monic polynomial over integers is an integer (cf. the rational root ...
The minimal polynomial f of α is unique.. To prove this, suppose that f and g are monic polynomials in J α of minimal degree n > 0. We have that r := f−g ∈ J α (because the latter is closed under addition/subtraction) and that m := deg(r) < n (because the polynomials are monic of the same degree).
This equation immediately gives the k-th Newton identity in k variables. Since this is an identity of symmetric polynomials (homogeneous) of degree k, its validity for any number of variables follows from its validity for k variables. Concretely, the identities in n < k variables can be deduced by setting k − n variables to zero.
It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity (/ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is