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The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and ...
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
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 term "algebraic equation" dates from the time when the main problem of algebra was to solve univariate polynomial equations. This problem was completely solved during the 19th century; see Fundamental theorem of algebra, Abel–Ruffini theorem and Galois theory. Since then, the scope of algebra has been dramatically enlarged.
In algebra, a multilinear polynomial [1] is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of 2 {\displaystyle 2} or higher; that is, each monomial is a constant times a product of distinct variables.
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
The problem of determining if a given set of Wang tiles can tile the plane. The problem of determining the Kolmogorov complexity of a string. Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers.