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In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression (including variables such as $a$ , $b$ and $c$ ).^[1] When the coefficients are themselves variables, they may also be called parameters.

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 of a polynomial, a series, or any expression. 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, the variables are often denoted by $x$ , $y$ , ..., and the parameters by $a$ , $b$ , $c$ , ..., but this is not always the case. For example, if $y$ is considered a parameter in the above expression, then the coefficient of $x$ would be $-3 y$ , and the constant coefficient (with respect to $x$ ) would be $1.5 + y$ .

When one writes $ax^{2}+bx+c,$ it is generally assumed that $x$ is the only variable, and that $a$ , $b$ and $c$ are parameters; thus the constant coefficient is $c$ in this case.

Any polynomial in a single variable $x$ 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 $a_{i}\neq 0$ (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.

Linear algebra

In linear algebra, a system of linear equations is frequently represented by its coefficient matrix. For example, the system of equations ${\begin{cases}2x+3y=0\\5x-4y=0\end{cases}},$ the associated coefficient matrix is ${\begin{pmatrix}2&3\\5&-4\end{pmatrix}}.$ 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^{[citation needed]}) of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrix ${\begin{pmatrix}1&2&0&6\\0&2&9&4\\0&0&0&4\\0&0&0&0\end{pmatrix}},$ 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 $v$ in a vector space with basis $\lbrace e_{1},e_{2},\dotsc ,e_{n}\rbrace$ are the coefficients of the basis vectors in the expression $v=x_{1}e_{1}+x_{2}e_{2}+\dotsb +x_{n}e_{n}.$

References

^ Weisstein, Eric W. "Coefficient". mathworld.wolfram.com. Retrieved 2020-08-15.

@@ Line 5: / Line 5: @@
 {{refimprove|date=May 2017}}
 }}
-In [[mathematics]], a '''coefficient''' is a multiplicative factor in some [[Summand|term]] of a [[polynomial]], a [[series (mathematics)|series]], or any [[expression (mathematics)|expression]]; it is usually a number, but may be any expression (including variables such as {{mvar|a}}, {{mvar|b}} and {{mvar|c}}).<ref name=":0">{{Cite web |title=Definition of Coefficient |url=https://www.mathsisfun.com/definitions/coefficient.html |access-date=2020-08-15 |website=www.mathsisfun.com}}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Coefficient |url=https://mathworld.wolfram.com/Coefficient.html |access-date=2020-08-15 |website=mathworld.wolfram.com |language=en}}</ref> When the coefficients are [[variable (mathematics)|variables]], they are often called [[parameter]]s.
+In [[mathematics]], a '''coefficient''' is a multiplicative factor in some [[Summand|term]] of a [[polynomial]], a [[series (mathematics)|series]], or any [[expression (mathematics)|expression]]; it is usually a number, but may be any expression (including variables such as {{mvar|a}}, {{mvar|b}} and {{mvar|c}}).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Coefficient |url=https://mathworld.wolfram.com/Coefficient.html |access-date=2020-08-15 |website=mathworld.wolfram.com |language=en}}</ref> When the coefficients are themselves [[variable (mathematics)|variables]], they may also be called [[parameter]]s.
+For example, the polynomial <math>2x^2-x+3</math> has coefficients 2, −1, and 3, and the powers of the variable <math>x</math> in the polynomial <math>ax^2+bx+c</math> have coefficient parameters <math>a</math>, <math>b</math>, and <math>c</math>.
-For example,
+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.
-<math>2x^2-x+3</math>, has the real coefficients 2, -1, and 3 respectively, and
+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 ==
-<math>ax^2+bx+c</math>, has coefficient parameters <math>a</math>, <math>b</math>, and <math>c</math> respectively, assuming <math>x</math> is the variable of the equation.
+In mathematics, a '''coefficient''' is a multiplicative factor in some [[Summand|term]] of a [[polynomial]], a [[series (mathematics)|series]], or any [[expression (mathematics)|expression]].  For example, in the polynomial
+<math display="block">7x^2-3xy+1.5+y,</math>
+with variables <math>x</math> and <math>y</math>, 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]], the variables are often denoted by {{mvar|x}}, {{mvar|y}}, ..., and the parameters by {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, ..., but this is not always the case. For example, if {{mvar|y}} is considered a parameter in the above expression, then the coefficient of {{mvar|x}} would be {{math|−3''y''}}, and the constant coefficient (with respect to {{mvar|x}}) would be {{math|1.5 + ''y''}}.
-The '''constant coefficient''' is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the real coefficient 3 and the parameter represented by ''c''.
-Similarly, the coefficient attached to the highest multiplicity 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 the parameter represented by ''a''.
-The [[binomial coefficient]]s occur in the expanded form of <math>(x+y)^n</math>, and are tabulated in [[Pascal's triangle]].
-== Terminology and Definition ==
-In mathematics, a '''coefficient''' is a multiplicative factor in some [[Summand|term]] of a [[polynomial]], a [[series (mathematics)|series]], or any [[expression (mathematics)|expression]];
-For example, in
-:<math>7x^2-3xy+1.5+y,</math>
-the first two terms have the coefficients 7 and −3, respectively. The third term 1.5 is a constant coefficient. The final term does not have any explicitly-written coefficient factor that would not change the term; the coefficient is thus taken to be 1 (since variables without number have a coefficient of 1).<ref name=":0" />
-In many scenarios, coefficients are numbers (as is the case for each term of the above 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]], the variables are often denoted by {{mvar|x}}, {{mvar|y}}, ..., and the parameters by {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, ..., but this is not always the case. For example, if {{mvar|y}} is considered a parameter in the above expression, then the coefficient of {{mvar|x}} would be {{math|−3''y''}}, and the constant coefficient (always with respect to {{mvar|x}}) would be {{math|1.5 + ''y''}}.
 When one writes
-:<math>ax^2+bx+c,</math>
+<math display="block">ax^2+bx+c,</math>
 it is generally assumed that {{mvar|x}} is the only variable, and that {{mvar|a}}, {{mvar|b}} and {{mvar|c}} are parameters; thus the constant coefficient is {{mvar|c}} in this case.
-{{anchor|leading coefficient}}Similarly, any [[polynomial]] in one variable {{mvar|x}} can be written as
+{{anchor|leading coefficient}}Any [[polynomial]] in a single variable {{mvar|x}} can be written as
-:<math>a_k x^k + \dotsb + a_1 x^1 + a_0</math>
+<math display="block">a_k x^k + \dotsb + a_1 x^1 + a_0</math>
-for some positive integer <math>k</math>, where <math>a_k, \dotsc, a_1, a_0</math> are coefficients; to allow this kind of expression in all cases, one must allow introducing terms with 0 as coefficient.
+for some [[nonnegative integer]] <math>k</math>, where <math>a_k, \dotsc, a_1, a_0</math> are the coefficients.  This includes the possibility that some terms have coefficient 0; for example, in <math>x^3 - 2x + 1</math>, the coefficient of <math>x^2</math> is 0, and the term <math>0x^2</math> does not appear explicitly.  For the largest <math>i</math> such that <math>a_i \ne 0</math> (if any), <math>a_i</math> is called the '''leading coefficient''' of the polynomial. For example, the leading coefficient of the polynomial
+<math display="block">4x^5 + x^3 + 2x^2</math>
-For the largest <math>i</math> with <math>a_i \ne 0</math> (if any), <math>a_i</math> is called the '''leading coefficient''' of the polynomial. For example, the leading coefficient of the polynomial
-:<math>\, 4x^5 + x^3 + 2x^2</math>
 is 4.
-Some specific coefficients that occur frequently in mathematics have dedicated names. For example, the [[binomial coefficient]]s occur in the expanded form of <math>(x+y)^n</math>, and are tabulated in [[Pascal's triangle]].
 ==Linear algebra==
-In [[linear algebra]], a [[system of linear equations]] is associated with a [[coefficient matrix]], which is used in [[Cramer's rule]] to find a solution to the system.
+In [[linear algebra]], a [[system of linear equations]] is frequently represented by its [[coefficient matrix]].  For example, the system of equations
+<math display="block">
+\begin{cases}
+x + 3y = 0 \\
+x - 4y = 0
+\end{cases},</math>
+the associated coefficient matrix is <math>\begin{pmatrix}
+& 3 \\
+& -4
+\end{pmatrix}.
+</math>  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, given the matrix described as follows:
+The '''leading entry''' (sometimes ''leading coefficient''{{cn}}) of a row in a matrix is the first nonzero entry in that row. So, for example, in the matrix
+<math display="block">
-:<math>
 \begin{pmatrix}
 & 2 & 0 & 6\\
@@ Line 56: / Line 51: @@
 \end{pmatrix},
 </math>
 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 [[constant (mathematics)|constants]] in elementary algebra, they can also be viewed as variables as the context broadens. For example, the [[coordinates]] <math>(x_1, x_2, \dotsc, x_n)</math> of a [[vector (geometric)|vector]] <math>v</math> in a [[vector space]] with [[basis (linear algebra)|basis]] <math>\lbrace e_1, e_2, \dotsc, e_n \rbrace </math>, are the coefficients of the basis vectors in the expression
+Though coefficients are frequently viewed as [[constant (mathematics)|constants]] in elementary algebra, they can also be viewed as variables as the context broadens. For example, the [[coordinates]] <math>(x_1, x_2, \dotsc, x_n)</math> of a [[vector (geometric)|vector]] <math>v</math> in a [[vector space]] with [[basis (linear algebra)|basis]] <math>\lbrace e_1, e_2, \dotsc, e_n \rbrace </math> are the coefficients of the basis vectors in the expression
-:<math> v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .</math>
+<math display="block"> v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .</math>
 ==See also==
@@ Line 66: / Line 60: @@
 *[[Degree of a polynomial]]
 *[[Monic polynomial]]
+*[[Binomial coefficient]]
 == References ==

Revision as of 13:54, 22 March 2022

Terminology and definition

Linear algebra

See also

References

Further reading