What is a Polynomial?
A polynomial is an expression built from variables and constants using only addition, subtraction, multiplication, and non-negative integer exponents. It sounds technical but you've been working with polynomials since you first learned algebra , every linear equation and quadratic is a polynomial.
In this lesson
1 What a Polynomial Is
A polynomial is a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power. Examples: 3x² + 2x − 7, x⁴ − 5x, 6 (a constant is a polynomial of degree 0), and 2x³ − x² + 4x + 1.
Polynomials can only have non-negative integer exponents (0, 1, 2, 3...). Terms like x⁻¹, x^(1/2), or 1/x are not polynomial terms.
2 Anatomy: Terms, Coefficients, Degree
In the polynomial 4x³ − 2x² + 7x − 9:
3 Types by Degree and Number of Terms
By degree: degree 0 = constant, degree 1 = linear (2x + 3), degree 2 = quadratic (x² + 5), degree 3 = cubic (x³ − 2), degree 4 = quartic, degree 5 = quintic.
By number of terms: 1 term = monomial (3x²), 2 terms = binomial (x + 4), 3 terms = trinomial (x² + 2x + 1), 4+ terms = polynomial (general).
4 Adding, Subtracting, and Multiplying
Adding/Subtracting: combine like terms (same variable, same exponent).
Multiplying: use the distributive property , multiply each term in the first polynomial by each term in the second.
5 What Is NOT a Polynomial
These are NOT polynomials and why: 1/x = x⁻¹ (negative exponent), √x = x^(1/2) (fractional exponent), 2ˣ (variable in the exponent , this is exponential, not polynomial), x/y (division by a variable).
2ˣ is NOT a polynomial , it's an exponential function. The variable must be the base, not the exponent. x² is polynomial; 2ˣ is not.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.