Introduction to Polynomials

Learn what polynomials are, how to identify their parts, and classify them by degree and number of terms.

Intermediate25 minLesson
Printable worksheet

Definition

A polynomial is an algebraic expression made up of one or more terms connected by addition or subtraction.
Each term is a product of:
  • A coefficient (a number)
  • Variables (like xx, yy), or none at all in a constant term
  • Raised to non-negative integer exponents
3x2+5x−73x^2 + 5x - 7
This polynomial has three terms:
  • 3x23x^2 (coefficient 3, variable xx, exponent 2)
  • 5x5x (coefficient 5, variable xx, exponent 1)
  • −7-7 (constant term, no variable)

Try it now

How many terms does the polynomial 3x2+5x−73x^2 + 5x - 7 have?

Worked Examples

For the polynomial 4x3−2x2+7x−54x^3 - 2x^2 + 7x - 5, identify the terms, coefficients, and degree.
1

List all terms

Separate by + and - signs → 4x34x^3, −2x2-2x^2, 7x7x, −5-5
2

Identify coefficients

The number in front of each variable → 4, -2, 7, -5
3

Find the degree of each term

The exponent on the variable → 3, 2, 1, 0
4

Find the degree of the polynomial

The highest degree among all terms → Degree = 3

Common Mistakes

Confusing the coefficient with the exponent

Why it's wrong: In 5x35x^3, students sometimes think 5 is the exponent. The coefficient is the number multiplying the variable; the exponent is the small number above.

Correct: In 5x35x^3: coefficient = 5, exponent = 3

Forgetting the coefficient of 1

Why it's wrong: When a term is written as x2x^2 instead of 1x21x^2, students forget there is a coefficient.

Correct: x2x^2 has a coefficient of 1. We just don't write it.

Thinking x\sqrt{x} or 1x\frac{1}{x} are polynomials

Why it's wrong: x=x1/2\sqrt{x} = x^{1/2} and 1x=x−1\frac{1}{x} = x^{-1} have fractional or negative exponents.

Correct: Polynomials only have whole number (non-negative integer) exponents: 0, 1, 2, 3, ...

Finding degree of a multi-variable polynomial incorrectly

Why it's wrong: For 3x2y33x^2y^3, the degree is the SUM of all exponents in that term: 2+3=52 + 3 = 5.

Correct: Add all exponents in each term, then find the highest total.

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

-4-2024-10-505101520xy

Tap or hover over the graph to read off values.

y = 3x + 7y = x² − 4x + 4y = 4x³ − 2x² + 7x − 5

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
How many terms does the polynomial 3x2+5x−73x^2 + 5x - 7 have?

Why It Matters

Polynomials are the building blocks of algebra and appear everywhere:
  • Physics: The path of a thrown ball follows a polynomial curve (h=−5t2+vt+h0h = -5t^2 + vt + h_0)
  • Economics: Revenue and cost functions are often polynomials
  • Engineering: Polynomials model structural stress, electrical circuits, and more
  • Computer Graphics: Curves in video games and animations use polynomials
Mastering polynomials opens doors to solving equations, graphing functions, and understanding advanced mathematics!

Real World Applications

Projectile Motion in Sports

When a basketball player shoots, the ball's height follows a polynomial equation.

Example:

The height hh in meters after tt seconds: h=−5t2+10t+2h = -5t^2 + 10t + 2. This is a degree 2 polynomial (quadratic).

1Try It Yourself

A soccer ball is kicked with height equation h=−4.9t2+15t+0.5h = -4.9t^2 + 15t + 0.5.

What is the degree of this polynomial, and what does each term represent?

Step 1: Write the mathematical expression

Identify the highest exponent:

Business Profit Modeling

Companies use polynomials to model costs, revenue, and profit based on units sold.

Example:

If profit is P=−0.5x2+20x−50P = -0.5x^2 + 20x - 50 where xx is hundreds of items sold, this polynomial helps find the optimal production level.

2Try It Yourself

A company's revenue is modeled by R=−2x2+100xR = -2x^2 + 100x dollars, where xx is the price in dollars.

Classify this polynomial by degree and number of terms.

Step 1: Choose the right answer

Count the terms and find the degree:

Key Takeaways

  • 1A polynomial is an expression with terms connected by + or -, where each term has non-negative integer exponents
  • 2Terms are the parts separated by + or - signs
  • 3The coefficient is the number multiplying the variable(s)
  • 4The degree of a term is the exponent (or sum of exponents for multiple variables)
  • 5The degree of a polynomial is the highest degree among all terms
  • 6Monomial = 1 term, Binomial = 2 terms, Trinomial = 3 terms
  • 7Standard form: terms arranged from highest to lowest degree

Frequently Asked Questions

Yes! A constant like 7 is a polynomial of degree 0. It's also called a constant polynomial or monomial.
Yes! A constant like 7 is a polynomial of degree 0. It's also called a constant polynomial or monomial.
All polynomials are expressions, but not all expressions are polynomials. Expressions like 1x\frac{1}{x} or x\sqrt{x} are NOT polynomials because they don't have non-negative integer exponents.
Yes! For example, 3x2y+5xy2−23x^2y + 5xy^2 - 2 is a polynomial in two variables (xx and yy). The degree of each term is the sum of all exponents.

Glossary

Polynomial
An algebraic expression with one or more terms, where each term has variables with non-negative integer exponents
Term
A single part of a polynomial (a number, variable, or their product)
Coefficient
The numerical factor multiplying a variable in a term
Degree (of a term)
The exponent of the variable, or sum of exponents if multiple variables
Degree (of polynomial)
The highest degree among all terms in the polynomial
Monomial
A polynomial with exactly one term
Binomial
A polynomial with exactly two terms
Trinomial
A polynomial with exactly three terms
Standard form
A polynomial written with terms in order from highest to lowest degree
Constant term
A term with no variable (degree 0)
Leading coefficient
The coefficient of the term with the highest degree

Formula Card

Polynomial Form

anxn+an−1xn−1+⋯+a1x+a0a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

Where $a_n, a_{n-1}, \ldots, a_0$ are coefficients and $n$ is a non-negative integer

Degree Classifications

Degree 0: Constant | Degree 1: Linear | Degree 2: Quadratic | Degree 3: Cubic | Degree 4: Quartic

Names based on the highest power of the variable

Today's challengeOne problem, three tries

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