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Chapter 8 (The solution of equations of the fifth degree at the Wayback Machine (archived 31 March 2010)) gives a description of the solution of solvable quintics x 5 + cx + d. Victor S. Adamchik and David J. Jeffrey, "Polynomial transformations of Tschirnhaus, Bring and Jerrard," ACM SIGSAM Bulletin, Vol. 37, No. 3, September 2003, pp. 90–94.
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.
The magic angle is a precisely defined angle, the value of which is approximately 54.7356°. The magic angle is a root of a second-order Legendre polynomial, P 2 (cos θ) = 0, and so any interaction which depends on this second-order Legendre polynomial vanishes at the magic angle.
Contributing structures of the carbonate ion. In chemistry, resonance, also called mesomerism, is a way of describing bonding in certain molecules or polyatomic ions by the combination of several contributing structures (or forms, [1] also variously known as resonance structures or canonical structures) into a resonance hybrid (or hybrid structure) in valence bond theory.
Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. [8] For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the ...
Clar's rule states that for a benzenoid polycyclic aromatic hydrocarbon (i.e. one with only hexagonal rings), the resonance structure with the largest number of disjoint aromatic π-sextets is the most important to characterize its chemical and physical properties. Such a resonance structure is called a Clar structure. In other words, a ...
Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (+) (), one can put it in standard form by expanding the products (by distributivity) and combining the like terms; for example, (+) = is of degree 1, even though each summand has ...
The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra , which does not provide any tool for computing exactly the solutions, although Newton's method allows ...