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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.
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
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 commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A.
The roots of the characteristic polynomial () are the eigenvalues of ().If there are n distinct eigenvalues , …,, then () is diagonalizable as () =, where D is the diagonal matrix and V is the Vandermonde matrix corresponding to the λ 's: = [], = [].
T 3 ⋅ e 1 = −4T 2 ⋅ e 1 − T ⋅ e 1 + e 1, so that: μ T, e 1 = X 3 + 4X 2 + X − I. This is in fact also the minimal polynomial μ T and the characteristic polynomial χ T : indeed μ T, e 1 divides μ T which divides χ T, and since the first and last are of degree 3 and all are monic, they must all
The polynomial factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order q. This implies that, if q = p n then X q − X is the product of all monic irreducible polynomials over GF(p), whose degree divides n.
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