Polynomial function in the context of Elementary function

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function ${\displaystyle f}$, is a member ${\displaystyle x}$ of the domain of ${\displaystyle f}$ such that ${\displaystyle f(x)}$ vanishes at ${\displaystyle x}$; that is, the function ${\displaystyle f}$ attains the value of 0 at ${\displaystyle x}$, or equivalently, ${\displaystyle x}$ is a solution to the equation ${\displaystyle f(x)=0}$. A "zero" of a function is thus an input value that produces an output of 0.

A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial ${\displaystyle f}$ of degree two, defined by ${\displaystyle f(x)=x^{2}-5x+6=(x-2)(x-3)}$ has the two roots (or zeros) that are 2 and 3.$${\displaystyle f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.}$$

In mathematics, a quadratic function of a single variable is a function of the form

where ⁠${\displaystyle x}$⁠ is its variable, and ⁠${\displaystyle a}$⁠, ⁠${\displaystyle b}$⁠, and ⁠${\displaystyle c}$⁠ are coefficients. The expression ⁠${\displaystyle \textstyle ax^{2}+bx+c}$⁠, especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two. In elementary mathematics a polynomial and its associated polynomial function are rarely distinguished and the terms quadratic function and quadratic polynomial are nearly synonymous and often abbreviated as quadratic.

In mathematics, the term linear function refers to two distinct but related notions:

• In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For distinguishing such a linear function from the other concept, the term affine function is often used.
• In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map.

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change (are invariant) under the transformations from a given linear group. For example, if we consider the action of the special linear group SL_n on the space of n by n matrices by left multiplication, then the determinant is an invariant of this action because the determinant of A X equals the determinant of X, when A is in SL_n.

The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved.

Here, "quickly" means an algorithm exists that solves the task and runs in polynomial time (as opposed to, say, exponential time), meaning the task completion time is bounded above by a polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is "P" or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be verified in polynomial time is "NP", standing for "nondeterministic polynomial time".

In algebra, a cubic equation in one variable is an equation of the form$${\displaystyle ax^{3}+bx^{2}+cx+d=0}$$in which a is not zero.

The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.

The set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K.

In mathematics, the term linear function refers to two distinct but related notions:

• In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero (a constant polynomial) or one (a linear polynomial). For distinguishing such a linear function from the other concept, the term affine function is often used.
• In linear algebra, mathematical analysis, and functional analysis, a linear function is a kind of function between vector spaces.