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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.
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 Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials (). Its existence can be shown using the notion of holonomic D-modules . Kashiwara (1976) proved that all roots of the Bernstein–Sato polynomial are negative rational numbers .
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 ...
Graph of the polynomial function x 4 + x 3 – x 2 – 7x/4 – 1/2 (in green) together with the graph of its resolvent cubic R 4 (y) (in red). The roots of both polynomials are visible too. In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:
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.