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  2. Geometrical properties of polynomial roots - Wikipedia

    en.wikipedia.org/wiki/Geometrical_properties_of...

    It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis. This can be extended to algebraic conjugation: the roots of a polynomial with rational coefficients are conjugate (that is, invariant) under the action of the Galois group of the polynomial. However, this symmetry can rarely be ...

  3. Rational root theorem - Wikipedia

    en.wikipedia.org/wiki/Rational_root_theorem

    q is an integer factor of the leading coefficient a n. The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n = 1.

  4. Eisenstein's theorem - Wikipedia

    en.wikipedia.org/wiki/Eisenstein's_theorem

    In mathematics, Eisenstein's theorem, named after the German mathematician Gotthold Eisenstein, applies to the coefficients of any power series which is an algebraic function with rational number coefficients. Through the theorem, it is readily demonstrable, for example, that the exponential function must be a transcendental function.

  5. Descartes' rule of signs - Wikipedia

    en.wikipedia.org/wiki/Descartes'_rule_of_signs

    In mathematics, Descartes' rule of signs, described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting zero coefficients), and the difference ...

  6. Transcendental number theory - Wikipedia

    en.wikipedia.org/wiki/Transcendental_number_theory

    Ultimately if it is possible to show that no finite degree or size of coefficient is sufficient then the number must be transcendental. Since a number α is transcendental if and only if P(α) ≠ 0 for every non-zero polynomial P with integer coefficients, this problem can be approached by trying to find lower bounds of the form

  7. Factorization of polynomials - Wikipedia

    en.wikipedia.org/wiki/Factorization_of_polynomials

    Square-free factorization; Factorization over finite fields; and reductions: From the multivariate case to the univariate case. From coefficients in a purely transcendental extension to the multivariate case over the ground field (see below). From coefficients in an algebraic extension to coefficients in the ground field (see below).

  8. Polynomial transformation - Wikipedia

    en.wikipedia.org/wiki/Polynomial_transformation

    Let = + + +be a polynomial, and , …, be its complex roots (not necessarily distinct). For any constant c, the polynomial whose roots are +, …, + is = = + + +.If the coefficients of P are integers and the constant = is a rational number, the coefficients of Q may be not integers, but the polynomial c n Q has integer coefficients and has the same roots as Q.

  9. Heaviside cover-up method - Wikipedia

    en.wikipedia.org/wiki/Heaviside_cover-up_method

    The Heaviside cover-up method, named after Oliver Heaviside, is a technique for quickly determining the coefficients when performing the partial-fraction expansion of a rational function in the case of linear factors. [1] [2] [3] [4]

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