Mathwords logoMathwords

Polynomial Term — Definition, Formula & Examples

A polynomial term is a single piece of a polynomial — a number, a variable, or a number multiplied by one or more variables raised to whole-number powers. For example, in the polynomial 3x2+5x−73x^2 + 5x - 7, the three terms are 3x23x^2, 5x5x, and −7-7.

A term of a polynomial is an expression of the form ax1n1x2n2⋯xknkax^{n_1}_1 x^{n_2}_2 \cdots x^{n_k}_k, where aa is a real-number coefficient and each nin_i is a non-negative integer exponent. Terms are separated by addition or subtraction within the polynomial.

Key Formula

axnax^n
Where:
  • aa = The coefficient (numerical factor) of the term
  • xx = The variable
  • nn = A non-negative integer exponent

How It Works

To identify the terms in a polynomial, look for the pieces separated by ++ or −- signs. Each term has two main parts: the coefficient (the numerical factor) and the variable part (the letters with their exponents). A term with no variable, like −7-7, is called a constant term. The degree of a term is the sum of all its variable exponents.

Worked Example

Problem: Identify every term in the polynomial 2x3−4x2+x−92x^3 - 4x^2 + x - 9, and state each term's coefficient and degree.
Separate the terms: Split the polynomial at each + or − sign, keeping the sign with the term that follows it.
2x3,−4x2,x,−92x^3,\quad -4x^2,\quad x,\quad -9
Find coefficients: Read the numerical factor of each term. Remember that xx by itself has a coefficient of 1.
2,−4,1,−92,\quad -4,\quad 1,\quad -9
Find degrees: The degree of each term is its variable's exponent. A constant has degree 0.
3,2,1,03,\quad 2,\quad 1,\quad 0
Answer: The polynomial has four terms: 2x32x^3 (coefficient 2, degree 3), −4x2-4x^2 (coefficient −4-4, degree 2), xx (coefficient 1, degree 1), and −9-9 (coefficient −9-9, degree 0).

Why It Matters

Recognizing individual terms is the first step in combining like terms, factoring, and finding the degree of a polynomial. These skills appear constantly in algebra courses and on standardized tests like the SAT.

Common Mistakes

Mistake: Forgetting that subtraction means the next term has a negative coefficient.
Correction: In 5x2−3x5x^2 - 3x, the second term is −3x-3x with coefficient −3-3, not 3x3x. Always attach the sign to the term that follows it.