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Constant term

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In mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial,

x^{2}+2x+3,\

The number 3 is a constant term.^[1]

After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial

ax^{2}+bx+c,\

where $x$ is the variable, as having a constant term of $c.$ If the constant term is 0, then it will conventionally be omitted when the quadratic is written out.

Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of $x^{0}.$ In particular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial

x^{2}+2xy+y^{2}-2x+2y-4\

has a constant term of −4, which can be considered to be the coefficient of $x^{0}y^{0},$ where the variables are eliminated by being exponentiated to 0 (any non-zero number exponentiated to 0 becomes 1). For any polynomial, the constant term can be obtained by substituting in 0 instead of each variable; thus, eliminating each variable. The concept of exponentiation to 0 can be applied to power series and other types of series, for example in this power series:

a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots ,

$a_{0}$ is the constant term.

Constant of integration

[edit]

The derivative of a constant term is 0, so when a term containing a constant term is differentiated, the constant term vanishes, regardless of its value. Therefore the antiderivative is only determined up to an unknown constant term, which is called "the constant of integration" and added in symbolic form (usually denoted as $C$ ).^[2]

For example, the antiderivative of $\cos x$ is $\sin x$ , since the derivative of $\sin x$ is equal to $\cos x$ based on the properties of trigonometric derivatives.

However, the integral of $\cos x$ is equal to $\sin x$ (the antiderivative), plus an arbitrary constant:

$\int \cos x\,\mathrm {d} x=\sin x+C,$

because for any constant $C$ , the derivative of the right-hand side of the equation is equal to the left-hand side of the equation.

References

[edit]

^ Fred Safier (2012). Schaum's Outline of Precalculus (3rd ed.). McGraw-Hill Education. p. 7. ISBN 978-0-07-179560-9.
^ Arthur Sherburne Hardy (1892). Elements of the Differential and Integral Calculus. Ginn & Company. p. 168.

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View on Grokipedia

from Grokipedia

In algebra and mathematics, a constant term refers to a component of a polynomial or algebraic expression that consists solely of a numerical value without any variables, such as 5 in the expression

3x^2 + 2x + 5

.^[1] This term remains fixed regardless of the values assigned to the variables, distinguishing it from variable terms like

ax

bx^2

.^[2] Constant terms play a fundamental role in defining the structure and behavior of polynomials, where they appear as the final term when the expression is written in standard form with descending powers of the variable, such as the

c

ax^2 + bx + c

.^[3] For instance, in a linear equation

y = mx + b

, the constant term

b

represents the y-intercept, shifting the graph vertically without affecting its slope.^[4] In higher-degree polynomials, like quadratics, the constant term influences the graph's position and the solutions to equations, appearing in the discriminant

b^2 - 4ac

.^[3] Beyond basic polynomials, constant terms are essential in more advanced contexts, such as series expansions or equations, where they provide the baseline value when variables are set to zero—for example, evaluating a polynomial at

x = 0

yields the constant term directly.^[4] They also ensure that polynomials are well-defined functions, with the constant term contributing to the overall degree classification only if it is the sole term (resulting in a degree-zero constant polynomial).^[2] Understanding constant terms is crucial for operations like addition, multiplication, and factoring, as they combine straightforwardly with other constants during simplification.^[1]

Definitions in Algebra

In Polynomials

In a polynomial expression, the constant term is defined as the term that contains no variables, equivalent to the coefficient of the degree-zero term.^[5] For a univariate polynomial written in standard form as

p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

, where each

a_i

is a constant coefficient, the constant term is

a_0

.^[1] This term represents a fixed numerical value independent of the variable

x

. Consider the polynomial

3x^2 + 2x - 5

; here, the constant term is -5, as it is the standalone numerical component after combining like terms.^[1] In multivariate polynomials, such as

4xy + 2x - 3y + 7

, the constant term is similarly the term without any variables, which is 7.^[5] The notation distinguishing constant terms from variable terms emerged in the late 16th century through the work of French mathematician François Viète, who pioneered the use of letters to represent both constants and unknowns in polynomial expressions, facilitating clearer algebraic manipulation.^[6] A key property of the constant term in polynomials is its invariance under substitution of the variable; regardless of the value assigned to

x

(or other variables), the constant term remains unchanged, preserving its fixed value within the expression.^[7]

In General Expressions

In any algebraic expression, the constant term refers to the numerical or fixed-value component that does not depend on the variables present, typically identified after fully expanding the expression into its simplest form.^[8] For instance, in the product

(x + 1)(x + 2)

, expansion yields

x^2 + 3x + 2

, where 2 is the constant term as it remains unchanged regardless of the value of

x

.^[9] This definition applies broadly to expressions involving sums, products, or more complex forms, where the constant is the portion independent of variable substitution.^[10] Examples extend to rational functions, such as

\frac{1}{x} + 2

, the constant term is 2 after considering the form, representing the part unaffected by the variable dependence.^[11] To identify the constant term, one collects all variable-independent parts during expansion of sums and products; for example, in a sum like

2x + (y + 4) - y

, simplification to

2x + 4

isolates 4 as the constant.^[9] This process involves distributing and combining like terms without relying on degree ordering specific to polynomials.^[8] While coefficients are fixed numerical multipliers attached to variable terms—such as the 3 in

3x

—the constant term specifically denotes the zero-degree or standalone fixed value in the expanded expression, distinguishing it as the intercept-like component when variables are set to zero.^[10] This differentiation ensures clarity in evaluating or manipulating general expressions.^[12]

Properties and Evaluation

Extraction Methods

One straightforward method for extracting the constant term from a univariate polynomial

p(x) = a_n x^n + \cdots + a_1 x + a_0

is direct substitution by evaluating

p(0)

, which isolates

a_0

since all higher-degree terms vanish. For instance, consider

p(x) = x^2 + 3x + 2

; substituting

x = 0

yields

p(0) = 2

, the constant term. This approach is particularly efficient for simple evaluation without needing to expand or manipulate the expression further. For products of binomials, such as

(a + b x)^n

, the binomial theorem provides a systematic way to identify the constant term by examining the general term

\binom{n}{k} a^{n-k} (b x)^k

, where the constant arises when

k = 0

, giving

a^n

. This term corresponds to the expansion's first component, free of the variable

x

. An adaptation of Horner's method, implemented via synthetic division with divisor

x

(root 0), allows extraction of the constant term as the remainder without fully evaluating higher coefficients unnecessarily.^[13] For a cubic polynomial

p(x) = 4x^3 + 5x^2 - 2x + 7

, set up synthetic division using 0:

Coefficients: 4 | 5 | -2 | 7
Bring down 4.
Multiply by 0: 4 × 0 = 0; add to 5: 5.
Multiply by 0: 5 × 0 = 0; add to -2: -2.
Multiply by 0: -2 × 0 = 0; add to 7: 7.

The remainder 7 is the constant term, with quotient coefficients 4, 5, -2.^[13] This variant leverages the nested structure of Horner's scheme for computational efficiency in coefficient isolation.^[13] In computational settings, computer algebra systems like SymPy facilitate constant term extraction through polynomial objects, where the trailing coefficient method retrieves the lowest-degree term's coefficient, equivalent to the constant for standard univariate forms.^[14]

Role in Function Evaluation

In polynomial functions, the constant term determines the value of the function at zero,

f(0)

, which serves as the y-intercept when graphing the function against the x-axis.^[15]^[16] For a general polynomial

p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0

, substituting

x = [0](/page/0)

yields

p([0](/page/0)) = a_0

, highlighting the constant term's direct role in establishing this baseline point on the graph.^[15] The constant term also represents the function's baseline value, independent of the input variables, providing insight into the inherent output without variable influence. In economic modeling, particularly in linear regression for cost functions, this term corresponds to fixed costs, which remain constant regardless of production levels; for example, in a total cost model

TC = \beta_0 + \beta_1 Q

, the intercept

\beta_0

captures fixed expenses like rent or salaries.^[17] The constant term influences the location and nature of a function's roots. If the constant term is zero, then

f(0) = 0

, making

x = 0

a root of the polynomial.^[18] In the quadratic equation

ax^2 + bx + c = 0

, a zero constant term

c = 0

implies one root at

x = 0

and the other at

x = -b/a

; more generally,

c

affects the discriminant

D = b^2 - 4ac

, which determines whether the roots are real and distinct, repeated, or complex.^[19] For multivariate functions, the constant term gives the value when all variables are zero, analogous to the univariate case. Consider

f(x, y) = xy + x + y + 1

; here, the constant term 1 equals

f(0, 0)

, representing the baseline independent of

x

and

y

.^[20]

Applications in Calculus

Constant of Integration

In calculus, the constant of integration arises in the computation of indefinite integrals, where the antiderivative of a function

f(x)

is expressed as

\int f(x) \, dx = F(x) + C

, with

C

representing an arbitrary constant term added to the particular antiderivative

F(x)

.^[21] This constant accounts for the fact that any antiderivative, when differentiated, yields the original function

f(x)

, as the derivative of a constant is zero.^[22] Thus, the indefinite integral produces a family of functions, all differing by constants, that satisfy the integration.^[23] The origin of this constant term stems from the inverse relationship between integration and differentiation: since differentiation eliminates additive constants, integration must introduce an arbitrary one to capture all possible solutions.^[21] For instance, if

F(x)

is an antiderivative of

f(x)

, then so is

F(x) + k

for any constant

k

, as

\frac{d}{dx}[F(x) + k] = f(x)

.^[22] This non-uniqueness ensures the general solution encompasses the complete set of functions whose derivatives match

f(x)

. Notation for the constant of integration typically uses

+C

+c

in single-variable cases, though variations like

+K

appear in some texts; in successive integrations, multiple distinct constants (e.g.,

+c_1 + c_2

) may be introduced before combining them into a single arbitrary constant.^[22] For multiple integrals, such as in vector calculus contexts, the constant can take the form of a vector or multiple components to account for the higher-dimensional family of solutions.^[22] Historically, the integral notation originated with Gottfried Wilhelm Leibniz in 1675, who introduced the

\int

symbol without explicitly denoting the constant, which evolved into the modern

+C

convention in 19th- and 20th-century textbooks as calculus formalized.^[24] To determine the specific value of

C

, initial or boundary conditions are applied to the general solution. For example, consider

\int x \, dx = \frac{1}{2}x^2 + C

; if the function satisfies

f(0) = 3

, substituting gives

3 = \frac{1}{2}(0)^2 + C

, so

C = 3

.^[22] This process fixes the constant, yielding the particular solution

f(x) = \frac{1}{2}x^2 + 3

, which is essential for solving initial value problems in applied contexts.^[23]

Taylor Series Expansion

In the Taylor series expansion of a function

f(x)

about a point

a

, the series is expressed as

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,

where the constant term corresponds to the

n=0

component, which is

f(a)

, representing the zeroth-order approximation or the function's value exactly at the expansion point

x = a

.^[25] This term serves as the foundational element of the polynomial approximation, ensuring that the series matches

f(x)

precisely when

x = a

.^[26] The role of the constant term is particularly prominent in providing the baseline value for approximations near

a

; for small deviations

|x - a|

, higher-order terms diminish in influence, making

f(a)

the dominant contributor to the function's behavior in that vicinity.^[25] In the special case of a Maclaurin series, where the expansion point is

a = [0](/page/0)

, the constant term simplifies directly to

f([0](/page/0))

, offering an immediate evaluation of the function at the origin without additional scaling.^[26] A classic example is the Taylor series for the exponential function

e^x

expanded about

a = 0

, given by

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots,

where the constant term is 1, equivalent to

e^0

.^[25] More generally, about an arbitrary point

a

, the expansion becomes

e^x = e^a \sum_{n=0}^{\infty} \frac{(x - a)^n}{n!}

, with

e^a

as the constant term.^[26] To compute the constant term, one evaluates the function itself at

a

, as it requires no derivatives, in contrast to higher terms that involve successive differentiations: the general

n

th coefficient is

\frac{f^{(n)}(a)}{n!}

, but for

n=0

, this reduces to the initial function evaluation

f(a)

.^[25] This simplicity underscores its role in initializing series approximations, which are widely applied in numerical methods and physics to model functions locally around a reference point, where the constant term establishes the zeroth-order accuracy.^[26]

Broader Mathematical Contexts

In Differential Equations

In ordinary differential equations (ODEs), constant terms play a crucial role in parameterizing the general solution, which encompasses all possible solutions to the equation. For a first-order linear ODE of the form

y' + p(x)y = q(x)

, the general solution obtained via the integrating factor method is

y(x) = e^{-\int p(x) \, dx} \left( \int q(x) e^{\int p(x) \, dx} \, dx + C \right)

, where

C

is an arbitrary constant that adjusts the solution to account for the infinite family of functions satisfying the equation.^[27] This constant arises from the indefinite integration process and represents the freedom in the solution space, distinguishing it from a particular solution that satisfies specific conditions.^[28] For higher-order linear ODEs, the general solution includes a number of arbitrary constants equal to the order of the equation. An

n

-th order linear ODE has a solution space of dimension

n

, requiring

n

independent constants to fully describe it.^[29] In the case of a second-order homogeneous linear ODE like

y'' + y = 0

, the general solution is

y(x) = A \cos x + B \sin x

, where

A

and

B

are arbitrary constants representing the contributions from the two linearly independent fundamental solutions.^[30] These constants embody the "constant term" components that scale the basis functions, allowing the solution to fit diverse initial or boundary scenarios. The arbitrary constants are determined by applying initial or boundary conditions to yield a unique particular solution. For an

n

-th order ODE,

n

such conditions are typically needed to fix the constants, transforming the general solution into one that satisfies the problem's constraints.^[29] This process highlights the distinction between the homogeneous solution, which solves the equation with zero right-hand side and contains all arbitrary constants, and the particular solution, which addresses the nonhomogeneous term without such freedom; the full general solution is their sum.^[31]

In Linear Algebra

In linear algebra, systems of linear equations are commonly expressed in matrix form as

Ax = b

, where

A

is the coefficient matrix,

x

is the column vector of unknown variables, and

b

is the column vector of constant terms that represent the inhomogeneous components of the system.^[32] These constant terms shift the solution set from the origin, distinguishing non-homogeneous systems from their homogeneous counterparts.^[33] To solve such systems, the augmented matrix

[A \mid b]

is formed by appending the constant vector

b

as the final column to the coefficient matrix

A

. For instance, the system

\begin{align*} 2x + y &= 3, \\ x + y &= 2 \end{align*}

has the augmented matrix

\begin{bmatrix} 2 & 1 & \mid & 3 \\ 1 & 1 & \mid & 2 \end{bmatrix},

where the constants 3 and 2 appear in the last column.^[34] Gaussian elimination is then applied to this augmented matrix to row-reduce it to row echelon form, revealing the system's consistency and isolating the contributions of the constants to the particular solution. In the example above, subtracting half the first row from the second yields

\begin{bmatrix} 2 & 1 & \mid & 3 \\ 0 & 0.5 & \mid & 0.5 \end{bmatrix},

and back-substitution gives the particular solution

x = 1

y = 1

; the constants ensure consistency here, as the rank of

A

matches the rank of

[A \mid b]

.^[35] Nonzero constants can lead to inconsistency if the ranks differ, rendering no solution possible.^[36] A homogeneous system arises when all constant terms are zero, i.e.,

Ax = 0

, resulting in the trivial solution

x = 0

and a solution space (the null space of

A

) whose dimension equals the nullity of

A

.^[37] In contrast, a non-homogeneous system

Ax = b

with

b \neq 0

has solutions forming an affine subspace: if consistent, the general solution is a particular solution

x_p

plus the homogeneous solutions, preserving the dimension of the solution space as that of the associated homogeneous system.^[38] The constants thus determine both solvability and the structure of the solution set.^[39]

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Constant term

Definitions in Algebra

In Polynomials

In General Expressions

Properties and Evaluation

Extraction Methods

Role in Function Evaluation

Applications in Calculus

Constant of Integration

Taylor Series Expansion

Broader Mathematical Contexts

In Differential Equations

In Linear Algebra

References

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See also

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Constant term

Definitions in Algebra

In Polynomials

In General Expressions

Properties and Evaluation

Extraction Methods

Role in Function Evaluation

Applications in Calculus

Constant of Integration

Taylor Series Expansion

Broader Mathematical Contexts

In Differential Equations

In Linear Algebra

References

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