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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.
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: = [], = [].
Applied to the monic polynomial + = with all coefficients a k considered as free parameters, this means that every symmetric polynomial expression S(x 1,...,x n) in its roots can be expressed instead as a polynomial expression P(a 1,...,a n) in terms of its coefficients only, in other words without requiring knowledge of the roots.
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).
In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A. The following three statements are equivalent: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...
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 a similar way, If f and g are two polynomial arithmetic functions, one defines f * g, the Dirichlet convolution of f and g, by () = () = = () where the sum extends over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m.
The Bernstein–Sato polynomial can also be defined for products of powers of several polynomials (Sabbah 1987). In this case it is a product of linear factors with rational coefficients. [citation needed] Nero Budur, Mircea Mustață, and Morihiko Saito generalized the Bernstein–Sato polynomial to arbitrary varieties.