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The graph of a degree 1 polynomial (or linear function) f(x) = a0 + a1x, where a1 ≠ 0, is an oblique line with y-intercept a0 and slope a1. The graph of a degree 2 polynomial. f(x) = a0 + a1x + a2x2, where a2 ≠ 0. is a parabola. The graph of a degree 3 polynomial. f(x) = a0 + a1x + a2x2 + a3x3, where a3 ≠ 0.
Fundamental theorem of algebra. The fundamental theorem of algebra, also called d'Alembert's theorem[1] or the d'Alembert–Gauss theorem, [2] states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex ...
The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free.
Curve fitting[ 1 ][ 2 ] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [ 3 ] possibly subject to constraints. [ 4 ][ 5 ] Curve fitting can involve either interpolation, [ 6 ][ 7 ] where an exact fit to the data is required, or smoothing, [ 8 ][ 9 ] in which a "smooth" function ...
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact that the graph of a polynomial in one ...
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
Definition. [edit] The Laplace operator is a second-order differential operator in the n -dimensional Euclidean space, defined as the divergence ( ) of the gradient ( ). Thus if is a twice-differentiable real-valued function, then the Laplacian of is the real-valued function defined by: Δ f f ∇ ∇ f. (1)