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In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum.
The declarative sentence is the most common kind of sentence in language, in most situations, and in a way can be considered the default function of a sentence. What this means essentially is that when a language modifies a sentence in order to form a question or give a command, the base form will always be the declarative.
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 ...
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 ...
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.
One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals (this was proven independently, using a similar method, by Niels Henrik Abel a few years before, and is the Abel–Ruffini theorem), and a systematic way for testing whether a specific polynomial ...
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography.
The polynomial x 2 + 1 = 0 has roots ± i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. The polynomial + has roots , +,, and thus can be factored as