Constant term
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 or 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 in 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](/page/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.