Classifying Polynomials

Learn to identify and classify polynomials by their degree and number of terms.

Intermediate20 minLesson
Printable worksheet

Definition

We classify polynomials in two ways:
By Number of Terms:
  • Monomial: 1 term (e.g., 5x25x^2, −3-3, 7xy7xy)
  • Binomial: 2 terms (e.g., x+5x + 5, 3x2−2x3x^2 - 2x)
  • Trinomial: 3 terms (e.g., x2+2x+1x^2 + 2x + 1)
  • Polynomial: 4+ terms (general term)
By Degree (highest exponent):
  • Constant: degree 0 (e.g., 77)
  • Linear: degree 1 (e.g., 3x+23x + 2)
  • Quadratic: degree 2 (e.g., x2−4x^2 - 4)
  • Cubic: degree 3 (e.g., x3+xx^3 + x)
  • Quartic: degree 4 (e.g., x4−1x^4 - 1)
Standard Form: Write terms from highest to lowest degree:
anxn+an−1xn−1+⋯+a1x+a0a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

Try it now

How many terms does 5x3−2x+75x^3 - 2x + 7 have?

Worked Examples

Classify each expression: (a) 4x34x^3, (b) 2x−72x - 7, (c) x2+3x−5x^2 + 3x - 5
1

Count terms in 4x34x^3

Only one term: 4x34x^3 → Monomial
2

Count terms in 2x−72x - 7

Two terms: 2x2x and −7-7 → Binomial
3

Count terms in x2+3x−5x^2 + 3x - 5

Three terms: x2x^2, 3x3x, and −5-5 → Trinomial

Common Mistakes

Counting −7-7 as having no degree

Why it's wrong: Every number can be written as −7x0-7x^0. Constants have degree 0, not "no degree."

Correct: Constants like −7-7 have degree 0 and are called constant polynomials.

Confusing degree with number of terms

Why it's wrong: A binomial can be any degree! x+1x + 1 (linear) and x5+3x^5 + 3 (quintic) are both binomials.

Correct: Degree = highest exponent. Number of terms = how many parts added/subtracted.

Forgetting to simplify before classifying

Why it's wrong: If x3x^3 and −x3-x^3 are both present, they cancel to zero!

Correct: Always combine like terms first, then classify the simplified form.

Not recognizing standard form

Why it's wrong: Standard form requires terms ordered from highest to lowest degree.

Correct: Rewrite 3+x2−2x3 + x^2 - 2x as x2−2x+3x^2 - 2x + 3 before classifying.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

Polynomial Basics Explained | Degree, Leading Term and Classification

by Understand The Math

Watch on YouTube

Interactive Visual

Function graph

-3-2-1012345-50510xy

Tap or hover over the graph to read off values.

53x + 1x² - 4x + 42x³ - x

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 3x + 2

m=3, b=2

Practice Problems

17 problems
Problem 1 of 17
Easy
How many terms does 5x3−2x+75x^3 - 2x + 7 have?

Why It Matters

Classifying polynomials helps you:
  • Communicate clearly: Saying "quadratic trinomial" immediately tells others you're working with something like x2+5x+6x^2 + 5x + 6
  • Choose solving methods: Different degrees require different techniques (factoring, quadratic formula, graphing)
  • Predict graph shapes: The degree tells you how many "turns" a graph can have
  • Organize your work: Standard form makes operations like addition much easier
In physics, economics, and engineering, recognizing polynomial types helps choose the right mathematical tools.

Real World Applications

Physics: Motion Equations

The position of a falling object follows a quadratic polynomial: height $= -16t^2 + v_0t + h_0$.

Example:

A ball thrown upward from 5 feet at 20 feet per second has height h=−16t2+20t+5h = -16t^2 + 20t + 5. This is a quadratic trinomial.

1Try It Yourself

A rocket's height is modeled by h=−5t2+30th = -5t^2 + 30t.

What type of polynomial is this?

Step 1: Choose the right answer

Count terms and find the highest degree:

Economics: Cost Functions

Companies model costs with polynomials. Linear for simple costs, quadratic when efficiency changes with scale.

Example:

If producing xx items costs C=50+3x+0.01x2C = 50 + 3x + 0.01x^2 dollars, this is a quadratic trinomial showing costs increase faster at high production.

2Try It Yourself

A bakery's daily cost is C=200+2.5xC = 200 + 2.5x dollars for xx cupcakes.

Classify this cost polynomial.

Step 1: Choose the right answer

Identify the type:

Computer Graphics: Curves

Cubic polynomials create smooth curves in animation and design software.

Example:

Bezier curves use cubic polynomials like P(t)=t3−3t2+3tP(t) = t^3 - 3t^2 + 3t to draw smooth paths in graphic design.

3Try It Yourself

An animation path follows y=t3−ty = t^3 - t.

What type of polynomial defines this path?

Step 1: Choose the right answer

Classify the polynomial:

Key Takeaways

  • 1Polynomials are classified by number of terms: monomial (1), binomial (2), trinomial (3), or polynomial (4+)
  • 2Polynomials are classified by degree: constant (0), linear (1), quadratic (2), cubic (3), quartic (4)
  • 3Standard form lists terms from highest to lowest degree
  • 4Always simplify before classifying (combine like terms first)
  • 5A full classification includes both: e.g., "quadratic trinomial"

Frequently Asked Questions

Yes! A constant like 55 is a monomial with degree 0. It has one term (5=5x05 = 5x^0).
Yes! A constant like 55 is a monomial with degree 0. It has one term (5=5x05 = 5x^0).
A trinomial is a specific type of polynomial with exactly 3 terms. 'Polynomial' can mean any expression with one or more terms.
No! Polynomials only have non-negative integer exponents. Since x−2=1x2x^{-2} = \frac{1}{x^2} has a negative exponent, it's not a polynomial.
Degree 5 is quintic, degree 6 is sextic (or hexic), and degree 7 is septic. Beyond that, we usually just say 'degree n polynomial.'

Glossary

Monomial
A polynomial with exactly one term (e.g., 5x25x^2)
Binomial
A polynomial with exactly two terms (e.g., x+3x + 3)
Trinomial
A polynomial with exactly three terms (e.g., x2+2x+1x^2 + 2x + 1)
Degree
The highest exponent of the variable in a polynomial
Linear
A polynomial of degree 1 (highest power is x1x^1)
Quadratic
A polynomial of degree 2 (highest power is x2x^2)
Cubic
A polynomial of degree 3 (highest power is x3x^3)
Standard form
Writing a polynomial with terms ordered from highest to lowest degree
Leading coefficient
The coefficient of the term with the highest degree
Today's challengeOne problem, three tries

More in This Topic

Introduction to Polynomials

Intermediate25 min

FOIL Method

Intermediate25 min

Adding and Subtracting Polynomials

Advanced25 min

Multiplying Polynomials

Intermediate25 min