Polynomial - More advanced examples of polynomials

Also in linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. In abstract algebra, one may define polynomials with coefficients in any ring. In knot theory the Alexander polynomial, the Jones ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication. Every polynomial in expanded form is a sum of terms (where subtraction is carried out by addition of negative numbers). Polynomials are classified by their degree and number of variables. The degree of a term in a polynomial is the su ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication (where whole number powers are viewed as repeated multiplication). Every polynomial in expanded form is a sum of terms ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form where the coefficients a0, ..., an are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f(x) = 0 is the x-axis. The graph of a degree 0 polynomial f(x) = a0 , where a0 ≠ 0, is a horizontal line with y-intercept a0 The graph of a degree 1 polynomial (or linear function) f(x) = a0 + a1x , whe ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g if there exists a polynomial q in R[X] such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotie ...

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One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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In multivariate calculus, polynomials in several variables play an important role. These are the simplest multivariate functions and can be defined using addition and multiplication alone. An example of a polynomial in the variables x, y, and z is The total degree of such a multivariate polynomial can be found by adding the exponents of the variables in every term, and taking the maximum. The above polynomial f(x,&# ...

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A root or zero of a polynomial f is a number ζ so that f(ζ) = 0. The fundamental theorem of algebra states that a polynomial of degree n over the complex numbers has exactly n complex roots (not necessarily distinct ones). Therefore a polynomial can be factorized as where each ζi i ...

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Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has a root: this is the statement of the fundamental theorem of algebra. There is a difference between approximating roots and finding concrete closed formulas for them. Formulas for the roots of polynom ...

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For given constants (i.e., numbers) a0, …, an in some field (possibly but not limited to R or C) with an non-zero, for n > 0, then a polynomial (function) of degree n is a function of the form More concisely, a polynomial function can be written in sigma notation as The constants a0, …, an are called the co ...

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is a horizontal line with y-intercept a0. The graph of a degree 1 polynomial function (or linear function) f(x) = a0 + a1x, where is an oblique line with y-intercept a0 and slope See also:

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The function is an example of a cubic function with leading coefficient −7 and constant coefficient 3. ...

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Polynomial - Polynomials and calculus. One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials a ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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