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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 nonzero coefficient of highest degree) is equal to 1.
That polynomial differs from the one defined here by a sign (), so it makes no difference for properties like having as roots the eigenvalues of ; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when is even.
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: = [], = [].
In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field GF(p m).This means that a polynomial F(X) of degree m with coefficients in GF(p) = Z/pZ is a primitive polynomial if it is monic and has a root α in GF(p m) such that {,,,,, …} is the entire field GF(p m).
Monic Cecconi-Botella (born 1936), French pianist, music educator and composer; Monic Hendrickx (born 1966), Dutch actress; Monic Pérez (born 1990), Puerto Rican model and Miss Universe contestant; Joseph de Monic (c.1650–1707), military officer and acting Governor of Newfoundland
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
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.
E.g.: x**2 + 3*x + 5 will be represented as [1, 3, 5] """ out = list (dividend) # Copy the dividend normalizer = divisor [0] for i in range (len (dividend)-len (divisor) + 1): # For general polynomial division (when polynomials are non-monic), # we need to normalize by dividing the coefficient with the divisor's first coefficient out [i ...