Indeterminate (variable) in the context of Polynomials

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. An example of a polynomial of a single indeterminate ${\displaystyle x}$ is ${\displaystyle x^{2}-4x+7}$. An example with three indeterminates is ${\displaystyle x^{3}+2xyz^{2}-yz+1}$.

Polynomials appear in many 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 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 numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.

In mathematics, to solve an equation is to find the solutions of an equation, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality.A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set.

An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions.

In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the polynomial ${\displaystyle P(x_{1},x_{2})=2x_{1}x_{2}+x_{1}^{3}+4}$ at ${\displaystyle x_{1}=2,x_{2}=3}$ consists of computing ${\displaystyle P(2,3)=2\cdot 2\cdot 3+2^{3}+4=24.}$ See also Polynomial ring § Polynomial evaluation

For evaluating the univariate polynomial ${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},}$ the most naive method would use ${\displaystyle n}$ multiplications to compute ${\displaystyle a_{n}x^{n}}$, use ${\displaystyle n-1}$ multiplications to compute ${\displaystyle a_{n-1}x^{n-1}}$ and so on for a total of ${\displaystyle {\tfrac {n(n+1)}{2}}}$ multiplications and ${\displaystyle n}$ additions.Using better methods, such as Horner's rule, this can be reduced to ${\displaystyle n}$ multiplications and ${\displaystyle n}$ additions. If some preprocessing is allowed, even more savings are possible.

In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).

A formal power series is a special kind of formal series, of the form $${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots ,}$$where the ${\displaystyle a_{n},}$ called coefficients, are numbers or, more generally, elements of some ring, and the ${\displaystyle x^{n}}$ are formal powers of the symbol ${\displaystyle x}$ that is called an indeterminate or, commonly, a variable. Hence, formal power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a formal power series may not represent a function of its variables. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.