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The eigenvalues of a 4×4 matrix are the roots of a quartic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation. An example arises in the Timoshenko-Rayleigh theory of beam bending. [10]
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points. + + + + = where a ≠ 0.
The cruciform curve, or cross curve is a quartic plane curve given by the equation = where a and b are two parameters determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a 2 x 2 + b 2 y 2 = 1, and is therefore a rational plane algebraic curve of genus zero.
On the other hand, a projective quartic surface is a surface in projective space P 3 of the same form, but now f is a homogeneous polynomial of 4 variables of degree 4, so for example (,,,) = + + + .
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 ...
The discriminant of the quartic polynomial x 4 + cx 2 + dx + e. The surface represents points (c, d, e) where the polynomial has a repeated root. The cuspidal edge corresponds to the polynomials with a triple root, and the self-intersection corresponds to the polynomials with two different repeated roots.
An example of a quintic whose roots cannot be expressed in terms of radicals is x 5 − x + 1 = 0. Some quintics may be solved in terms of radicals. However, the solution is generally too complicated to be used in practice. Instead, numerical approximations are calculated using a root-finding algorithm for polynomials.
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 polynomial or to its terms. For example, the term 2x in x 2 + 2x + 1 is a linear term in a quadratic polynomial.