A polynomial is an expression of finite length constructed from variables (also called
indeterminates) and constants, using only the operations of addition, subtraction,
multiplication, and non-negative integer exponents. However, the division by a constant is
allowed, because the multiplicative inverse of a non zero constant is also a constant. For
example, x2 − x/4 + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term
involves division by the variable x (4/x), and also because its third term contains an exponent
that is not a non-negative integer (3/2).
The term "polynomial" can also be used as an adjective, for quantities that can be expressed as
a polynomial of some parameter, as in polynomial time, which is used in computational
complexity theory. Polynomial comes from the Greek poly, "many" and medieval Latin
binomium, "binomial". The word was introduced in Latin by Franciscus Vieta. Polynomials
appear in a wide variety of areas of mathematics and science. For example, they are used to
form polynomial equations, which encode a wide range of problems, from elementary word
problems to complicated problems in the sciences.
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Polynomial algorithms are at the core of classical "computer algebra". Incorporating methods
that span from antiquity to the latest cutting-edge research at Wolfram Research, Mathematica
has the world's broadest and deepest integrated web of polynomial algorithms. Carefully tuned
strategies automatically select optimal algorithms, allowing large-scale polynomial algebra to
become a routine part of many types of computations.
Algebraic Operations on Polynomials ;- For many kinds of practical calculations, the only
operations you will need to perform on polynomials are essentially structural ones. If you do
more advanced algebra with polynomials, however, you will have to use the algebraic
operations discussed in this tutorial. You should realize that most of the operations discussed
in this tutorial work only on ordinary polynomials, with integer exponents and rational-
number coefficients for each term.
Transforming Algebraic Expressions ;- There are often many different ways to write the
same algebraic expression. As one example, the expression can be written as . Mathematica
provides a large collection of functions for converting between different forms of algebraic
Polynomials Modulo Primes ;- Mathematica can work with polynomials whose coefficients
are in the finite field of integers modulo a prime .
Polynomials over Algebraic Number Fields ;- Functions like Factor usually assume that all
coefficients in the polynomials they produce must involve only rational numbers. But by
setting the option Extension you can extend the domain of coefficients that will be allowed.
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