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In mathematics, Gegenbauer polynomials or ultraspherical polynomials C (α) n (x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x 2) α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.
Here is an example of polynomial division as described above. Let: = +() = +P(x) will be divided by Q(x) using Ruffini's rule.The main problem is that Q(x) is not a binomial of the form x − r, but rather x + r.
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
Let p and q be polynomials with coefficients in an integral domain F, typically a field or the integers. A greatest common divisor of p and q is a polynomial d that divides p and q, and such that every common divisor of p and q also divides d. Every pair of polynomials (not both zero) has a GCD if and only if F is a unique factorization domain.
Enjoy a classic game of Hearts and watch out for the Queen of Spades!
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues , Sir James Ivory and Carl Gustav Jacobi .
The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first n power sums also form an algebraic basis for the space of symmetric polynomials.