Coefficient: Difference between revisions
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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}}). |
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. |
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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>. |
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For example, |
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<math>2x^2-x+3</math>, has the real coefficients 2, -1, and 3 respectively, and |
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<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. |
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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. |
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⚫ | 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''}}. |
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The [[binomial coefficient]]s occur in the expanded form of <math>(x+y)^n</math>, and are tabulated in [[Pascal's triangle]]. |
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For example, in |
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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" /> |
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⚫ | In many scenarios, coefficients are numbers (as is the case for each term of the |
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When one writes |
When one writes |
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<math display="block">ax^2+bx+c,</math> |
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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. |
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. |
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{{anchor|leading coefficient}} |
{{anchor|leading coefficient}}Any [[polynomial]] in a single variable {{mvar|x}} can be written as |
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<math display="block">a_k x^k + \dotsb + a_1 x^1 + a_0</math> |
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for some |
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 |
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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 |
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is 4. |
is 4. |
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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]]. |
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==Linear algebra== |
==Linear algebra== |
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In [[linear algebra]], a [[system of linear equations]] is |
In [[linear algebra]], a [[system of linear equations]] is frequently represented by its [[coefficient matrix]]. For example, the system of equations |
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<math display="block"> |
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\begin{cases} |
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2x + 3y = 0 \\ |
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5x - 4y = 0 |
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\end{cases},</math> |
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the associated coefficient matrix is <math>\begin{pmatrix} |
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2 & 3 \\ |
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5 & -4 |
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\end{pmatrix}. |
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</math> Coefficient matrices are used in algorithms such as [[Gaussian elimination]] and [[Cramer's rule]] to find solutions to the system. |
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The '''leading entry''' (sometimes ''leading coefficient'') of a row in a matrix is the first nonzero entry in that row. So, for example, |
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 |
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<math display="block"> |
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:<math> |
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\begin{pmatrix} |
\begin{pmatrix} |
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1 & 2 & 0 & 6\\ |
1 & 2 & 0 & 6\\ |
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\end{pmatrix}, |
\end{pmatrix}, |
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</math> |
</math> |
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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. |
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. |
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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> |
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 |
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<math display="block"> v = x_1 e_1 + x_2 e_2 + \dotsb + x_n e_n .</math> |
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==See also== |
==See also== |
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*[[Degree of a polynomial]] |
*[[Degree of a polynomial]] |
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*[[Monic polynomial]] |
*[[Monic polynomial]] |
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*[[Binomial coefficient]] |
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== References == |
== References == |
Revision as of 13:54, 22 March 2022
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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 has coefficients 2, −1, and 3, and the powers of the variable in the polynomial have coefficient parameters , , and .
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 with variables and , 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 −3y, and the constant coefficient (with respect to x) would be 1.5 + y.
When one writes 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 for some nonnegative integer , where are the coefficients. This includes the possibility that some terms have coefficient 0; for example, in , the coefficient of is 0, and the term does not appear explicitly. For the largest such that (if any), is called the leading coefficient of the polynomial. For example, the leading coefficient of the polynomial 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 the associated coefficient matrix is 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 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 of a vector in a vector space with basis are the coefficients of the basis vectors in the expression
See also
References
- ^ Weisstein, Eric W. "Coefficient". mathworld.wolfram.com. Retrieved 2020-08-15.
Further reading
- Sabah Al-hadad and C.H. Scott (1979) College Algebra with Applications, page 42, Winthrop Publishers, Cambridge Massachusetts ISBN 0-87626-140-3 .
- 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 .