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In mathematics, a coefficient is a multiplicative factor in some
term Term may refer to: * Terminology, or term, a noun or compound word used in a specific context, in particular: **Technical term, part of the specialized vocabulary of a particular field, specifically: ***Scientific terminology, terms used by scient ...
of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
, a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
, or an
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, o ...
; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves variables, they may also be called
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter ''c'', respectively. The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and ''a'', respectively.


Terminology and definition

In mathematics, a coefficient is a multiplicative factor in some
term Term may refer to: * Terminology, or term, a noun or compound word used in a specific context, in particular: **Technical term, part of the specialized vocabulary of a particular field, specifically: ***Scientific terminology, terms used by scient ...
of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
, a
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
, or any
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, o ...
. For example, in the polynomial 7x^2-3xy+1.5+y, with variables x and y, the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient. In the final term, the coefficient is 1 and is not explicitly written. In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. Following
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
, the variables are often denoted by , , ..., and the parameters by , , , ..., but this is not always the case. For example, if is considered a parameter in the above expression, then the coefficient of would be , and the constant coefficient (with respect to ) would be . When one writes ax^2+bx+c, it is generally assumed that is the only variable, and that , and are parameters; thus the constant coefficient is in this case. Any
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
in a single variable can be written as a_k x^k + \dotsb + a_1 x^1 + a_0 for some nonnegative integer k, where a_k, \dotsc, a_1, a_0 are the coefficients. This includes the possibility that some terms have coefficient 0; for example, in x^3 - 2x + 1, the coefficient of x^2 is 0, and the term 0x^2 does not appear explicitly. For the largest i such that a_i \ne 0 (if any), a_i is called the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial 4x^5 + x^3 + 2x^2 is 4. This can be generalised to multivariate polynomials with respect to a monomial order, see .


Linear algebra

In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
, a system of linear equations is frequently represented by its
coefficient matrix In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations. Coefficient matrix In general, a system with ''m'' linear ...
. For example, the system of equations \begin 2x + 3y = 0 \\ 5x - 4y = 0 \end, the associated coefficient matrix is \begin 2 & 3 \\ 5 & -4 \end. Coefficient matrices are used in algorithms such as Gaussian elimination and Cramer's rule to find solutions to the system. The leading entry (sometimes ''leading coefficient'') of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrix \begin 1 & 2 & 0 & 6\\ 0 & 2 & 9 & 4\\ 0 & 0 & 0 & 4\\ 0 & 0 & 0 & 0 \end, the leading coefficient of the first row is 1; that of the second row is 2; that of the third row is 4, while the last row does not have a leading coefficient. Though coefficients are frequently viewed as constants in elementary algebra, they can also be viewed as variables as the context broadens. For example, the coordinates (x_1, x_2, \dotsc, x_n) of a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
v in a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
with
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
\lbrace e_1, e_2, \dotsc, e_n \rbrace are the coefficients of the basis vectors in the expression v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .


See also

*
Correlation coefficient A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
* Degree of a polynomial *
Monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\c ...
* Binomial coefficient


References


Further reading

*Sabah Al-hadad and C.H. Scott (1979) ''College Algebra with Applications'', page 42, Winthrop Publishers, Cambridge Massachusetts . *Gordon Fuller, Walter L Wilson, Henry C Miller, (1982) ''College Algebra'', 5th edition, page 24, Brooks/Cole Publishing, Monterey California {{ISBN, 0-534-01138-1 . Polynomials Mathematical terminology Algebra Numbers Variables (mathematics)