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Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line; Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola.
The following names are assigned to polynomials according to their degree: [2] [3] [4] Special case – zero (see § Degree of the zero polynomial, below) Degree 0 – non-zero constant [5] Degree 1 – linear; Degree 2 – quadratic; Degree 3 – cubic; Degree 4 – quartic (or, if all terms have even degree, biquadratic) Degree 5 – quintic
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
Hill's equation is an important example in the understanding of periodic differential equations. Depending on the exact shape of f ( t ) {\displaystyle f(t)} , solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially. [ 3 ]
A periodic motion is a closed curve in phase space. That is, for some period, ′ = (,), = (). The textbook example of a periodic motion is the undamped pendulum.. If the phase space is periodic in one or more coordinates, say () = (+), with a vector [clarification needed], then there is a second kind of periodic motions defined by
In the mathematical field of numerical analysis, Runge's phenomenon (German:) is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points.
This problem is commonly resolved by the use of spline interpolation. Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree. Interpolation of periodic functions by harmonic functions is accomplished by Fourier transform.