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In mathematics, a cubic function is a function of the form () = + + +, that is, a polynomial function of degree three. In many texts, the coefficients a , b , c , and d are supposed to be real numbers , and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to ...
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 graph of a polynomial function of degree 3. The x occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is "indeterminate").
A primitive polynomial of degree m has m different roots in GF(p m), which all have order p m − 1, meaning that any of them generates the multiplicative group of the field. Over GF(p) there are exactly φ(p m − 1) primitive elements and φ(p m − 1) / m primitive polynomials, each of degree m, where φ is Euler's totient function. [1]
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 .
If the polynomial pieces P i each have degree at most n, then the spline is said to be of degree ≤ n (or of order n + 1). If S ∈ C r i {\displaystyle S\in C^{r_{i}}} in a neighborhood of t i , then the spline is said to be of smoothness (at least) C r i {\displaystyle C^{r_{i}}} at t i .
which is exact for polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1].
The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C([a, b]) of all continuous functions on [a, b] to itself. The map X is linear and it is a projection on the subspace () of polynomials of degree n or less. The Lebesgue constant L is defined as the operator norm of X.