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  2. Degree of a polynomial - Wikipedia

    en.wikipedia.org/wiki/Degree_of_a_polynomial

    For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...

  3. Abel–Ruffini theorem - Wikipedia

    en.wikipedia.org/wiki/Abel–Ruffini_theorem

    With modern computers and programs, deciding whether a polynomial is solvable by radicals can be done for polynomials of degree greater than 100. [6] Computing the solutions in radicals of solvable polynomials requires huge computations. Even for the degree five, the expression of the solutions is so huge that it has no practical interest.

  4. Quintic function - Wikipedia

    en.wikipedia.org/wiki/Quintic_function

    If a and b are rational numbers, the equation x 5 + ax + b = 0 is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ℓ and m such that

  5. Polynomial - Wikipedia

    en.wikipedia.org/wiki/Polynomial

    The polynomial 3x 2 − 5x + 4 is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two. [11]

  6. Hilbert's seventeenth problem - Wikipedia

    en.wikipedia.org/wiki/Hilbert's_seventeenth_problem

    Furthermore, if the polynomial has a degree 2d greater than two, there are significantly many more non-negative polynomials that cannot be expressed as sums of squares. [4] The following table summarizes in which cases every non-negative homogeneous polynomial (or a polynomial of even degree) can be represented as a sum of squares:

  7. Cubic equation - Wikipedia

    en.wikipedia.org/wiki/Cubic_equation

    This works well for every degree, but, in degrees higher than four, the resulting polynomial that has the s i as roots has a degree higher than that of the initial polynomial, and is therefore unhelpful for solving. This is the reason for which Lagrange's method fails in degrees five and higher.

  8. Galois theory - Wikipedia

    en.wikipedia.org/wiki/Galois_theory

    For the polynomial f(x) = x 5 − x − 1, the lone real root x = 1.1673... is algebraic, but not expressible in terms of radicals. The other four roots are complex numbers. Van der Waerden [11] cites the polynomial f(x) = x 5 − x − 1. By the rational root theorem, this has no rational zeroes. Neither does it have linear factors modulo 2 or 3.

  9. Sturm series - Wikipedia

    en.wikipedia.org/wiki/Sturm_series

    Let and two univariate polynomials. Suppose that they do not have a common root and the degree of p 0 {\displaystyle p_{0}} is greater than the degree of p 1 {\displaystyle p_{1}} . The Sturm series is constructed by: