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This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula.
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers) variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).
Two to the power of n, written as 2 n, is the number of values in which the bits in a binary word of length n can be set, where each bit is either of two values. A word, interpreted as representing an integer in a range starting at zero, referred to as an "unsigned integer", can represent values from 0 (000...000 2) to 2 n − 1 (111...111 2) inclusively.
Terms are within the same expression and are combined by either addition or subtraction. For example, take the expression: + There are two terms in this expression. Notice that the two terms have a common factor, that is, both terms have an . This means that the common factor variable can be factored out, resulting in
Lexicographic order (lex) first compares exponents of x 1 in the monomials, and in case of equality compares exponents of x 2, and so forth. The name is derived from the similarity with the usual alphabetical order used in lexicography for dictionaries, if monomials are represented by the sequence of the exponents of the indeterminates.
In this case, the two factors of −z 3 are coprime. This implies that three does not divide u and that the two factors are cubes of two smaller numbers, r and s. 2u = r 3 u 2 + 3v 2 = s 3. Since u 2 + 3v 2 is odd, so is s. A crucial lemma shows that if s is odd and if it satisfies an equation s 3 = u 2 + 3v 2, then it can be written in terms ...