Mathwords logoMathwords

Constant Term

Constant Term

The term in a simplified algebraic expression or equation which contains no variable(s). If there is no such term, the constant term is 0.

 

Example:   –5 is the constant term in p(x) = 2x3 – 4x2 + 9x – 5

 

See also

Leading term, leading coefficient

Key Formula

p(x)=anxn+an−1xn−1+⋯+a1x+a0p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0
Where:
  • a0a_0 = The constant term — the term with no variable attached
  • ana_n = The leading coefficient (coefficient of the highest-degree term)
  • xx = The variable
  • nn = The degree of the polynomial

Worked Example

Problem: Identify the constant term in the polynomial f(x)=5x3−2x2+8x−11f(x) = 5x^3 - 2x^2 + 8x - 11.
Step 1: List every term in the polynomial.
5x3,−2x2,8x,−115x^3, \quad -2x^2, \quad 8x, \quad -11
Step 2: Check which term has no variable. The terms 5x35x^3, −2x2-2x^2, and 8x8x all contain xx. The term −11-11 does not contain any variable.
Step 3: Equivalently, evaluate the polynomial at x=0x = 0. Every term with xx vanishes, leaving only the constant term.
f(0)=5(0)3−2(0)2+8(0)−11=−11f(0) = 5(0)^3 - 2(0)^2 + 8(0) - 11 = -11
Answer: The constant term is −11-11.

Another Example

This example covers the edge case where no explicit constant appears in the expression, so the constant term defaults to 0.

Problem: Find the constant term of g(x)=6x4+3x2−xg(x) = 6x^4 + 3x^2 - x after simplifying.
Step 1: List the terms of the polynomial.
6x4,3x2,−x6x^4, \quad 3x^2, \quad -x
Step 2: Every term contains the variable xx. There is no standalone number without a variable.
Step 3: By definition, when no explicit constant term appears, the constant term is 00. You can verify: g(0)=6(0)4+3(0)2−0=0g(0) = 6(0)^4 + 3(0)^2 - 0 = 0.
g(0)=0g(0) = 0
Answer: The constant term is 00.

Frequently Asked Questions

How do you find the constant term of a polynomial?
Look for the term that has no variable. Alternatively, substitute x=0x = 0 (or whatever the variable is) into the polynomial. Every term containing a variable becomes zero, and the value that remains is the constant term.
What is the difference between a constant term and a coefficient?
A coefficient is the numerical factor multiplied by a variable (for example, 77 in 7x27x^2). The constant term is the term that stands alone with no variable at all. You can think of the constant term as the coefficient of x0x^0, since x0=1x^0 = 1, but in everyday usage the two ideas serve different purposes.
Can the constant term be negative or a fraction?
Yes. The constant term can be any real number — positive, negative, zero, a fraction, or even an irrational number. For instance, in x2+34x^2 + \frac{3}{4}, the constant term is 34\frac{3}{4}.

Constant Term vs. Leading Term

Constant TermLeading Term
DefinitionThe term with no variable (degree 0)The term with the highest degree
Position in standard formLast term when written in descending orderFirst term when written in descending order
Example in 4x3−x+74x^3 - x + 7774x34x^3
What it tells youThe y-intercept of the polynomial's graphThe end behavior of the polynomial's graph

Why It Matters

The constant term gives you the y-intercept of a polynomial function immediately: when x=0x = 0, f(0)f(0) equals the constant term, so the graph crosses the y-axis at that value. In factoring, the constant term plays a key role in the Rational Root Theorem, which says every possible rational root is a factor of the constant term divided by a factor of the leading coefficient. You will also encounter constant terms when completing the square, working with quadratic formulas, and analyzing series expansions.

Common Mistakes

Mistake: Confusing the constant term's sign when it is subtracted.
Correction: In x2+3x−9x^2 + 3x - 9, the constant term is −9-9, not 99. Always include the sign that is attached to the term.
Mistake: Identifying the constant term before simplifying the expression.
Correction: If the expression is x2+4+2x−7x^2 + 4 + 2x - 7, first combine like terms to get x2+2x−3x^2 + 2x - 3. The constant term is −3-3, not 44 or −7-7 separately.

Related Terms

  • Term — A constant term is one specific type of term
  • Leading Term — The highest-degree term, opposite end from constant
  • Leading Coefficient — Coefficient of the leading term
  • Constant — A fixed value; the constant term is one example
  • Variable — The constant term is defined by having no variable
  • Expression — Constant terms appear inside algebraic expressions
  • Simplify — Must simplify before identifying the constant term
  • Algebra — Broader field where constant terms are used