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Monomials, Binomials, Polynomials

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Monomials, Binomials, Polynomials

Study Guide

Key Definition

A monomial is a single algebraic term: a product of numbers (including constants, e.g., 7 = 7·x^0) and variables raised to nonnegative integer exponents, such as $3xy$ or $-4x^2$. A binomial is the sum or difference of exactly two monomials, for example $x + 3$ or $x - 4$. A polynomial is an expression that is the sum or difference of one or more monomials (so monomials and binomials are special cases of polynomials).

Important Notes

Monomials are single terms (including constants like 7 = 7·x^0) and do not contain addition or subtraction between terms.
The coefficient of a monomial is its numerical factor.
A binomial has exactly two monomial terms joined by + or -.
A polynomial is an expression composed of one or more monomial terms; binomials and monomials are special cases.
Variables in monomials have nonnegative integer exponents.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$x^n$ represents a variable raised to the power $n$

$·$ or $×$ represents multiplication

$(…)$ indicates grouping via parentheses

Remember to use proper notation when solving problems

Why It Works

Understanding the structure of monomials, binomials, and polynomials allows you to simplify and solve algebraic expressions efficiently.

Remember

When multiplying monomials, add exponents; when dividing, subtract exponents. Count the number of terms to classify expressions: one term → monomial, two terms → binomial, three or more → polynomial.

Quick Reference

Monomial Example:$4x^3$

Binomial Example:$2x + 3$

Polynomial Example:$x^2 + 2x + 1$

Understanding Monomials, Binomials, Polynomials

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A monomial is a single algebraic term. An example is $5x^2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Identify the monomial from the following options.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are creating a garden with a rectangular area. The length is represented by $2x$ and the width by $x + 3$. Calculate the polynomial expression for the area.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the polynomial $x^2 + 5x + 6$. Factor it into two binomials.

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following expressions is a polynomial?

Recap

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Review key concepts and takeaways