Types of Polynomials — Classification with Examples

Polynomials are classified in two independent ways: by degree (constant, linear, quadratic, cubic, quartic, quintic, and higher) and by the number of terms (monomial, binomial, trinomial, and beyond). This article walks through every type, gives three worked examples spanning Quick to Stretch, lists the mistakes that cost marks, and credits the people who built the classification.

Classification by Degree

The degree of a polynomial is the highest exponent of the variable. Standard names by degree:

The zero polynomial (p(x)=0) has no degree — or, depending on convention, is assigned degree −∞. It is the only polynomial without a leading term.

Special degree categories:

• Constant polynomial — degree 0. Just a number with no variable, like 5 or -2.
• Zero polynomial — the polynomial that is identically zero. Distinct from a constant polynomial.

Classification by Number of Terms

A term in a polynomial is a single algebraic chunk separated by a + or − sign. Counting terms gives the second classification:

Note that the term-count classification only applies after like terms are combined.

The Two-Word Combined Label

Every polynomial carries both labels simultaneously. Some examples:

• 777 — constant monomial (degree 0, 1 term)
• 3x — linear monomial
• x + 2 — linear binomial
• x² - 9 — quadratic binomial
• x² - 5x + 6 — quadratic trinomial
• 2x³ — cubic monomial
• x³ + 1 — cubic binomial
• x³ - 6x² + 11x - 6 — cubic polynomial (4 terms, no special name)

Why Classifying Polynomials Matters

The point of classifying polynomials isn't pedagogy for its own sake — it tells you which solution method to reach for.

• Linear polynomials are solved by isolating the variable.
• Quadratic polynomials have the quadratic formula, factoring, or completing the square.
• Cubic and quartic polynomials have closed-form solutions but are usually factored using the Rational Root Theorem.
• Quintic and higher polynomials have no general closed-form solution.

The Slip-Ups That Cost Marks on Classifying Polynomials

Mistake 1: Classifying before combining like terms.

Where it slips in: any polynomial where like terms appear separately in the original expression.

Mistake 2: Treating degree and term-count as competing categories.

Where it slips in: students who think "cubic" and "trinomial" can't both describe the same polynomial.

Mistake 3: Calling a non-polynomial a "polynomial".

Where it slips in: expressions with negative or fractional exponents on the variable.

Polynomial Classification Master Table — Degree × Terms

Frequently Asked Questions

What are the types of polynomials by degree?
Constant (0), linear (1), quadratic (2), cubic (3), quartic (4), quintic (5), and higher — each named for its highest exponent.

What are the types of polynomials by number of terms?
Monomial (1 term), binomial (2 terms), trinomial (3 terms). Polynomials with 4 or more terms don't have a special name — just called "polynomial".

Is x\sqrt{x} a polynomial?
No. Polynomials require non-negative integer exponents on every variable.