What is a Monomial?

S6-SA1-0026

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A monomial is a type of algebraic expression that has only one term. This term can be a number, a variable, or a product of numbers and variables, where variables have only non-negative integer exponents.

Simple Example

Quick Example

Imagine you buy 'x' number of ladoos, and each ladoo costs 10 rupees. The total cost would be 10x. Here, '10x' is a monomial because it's a single term representing the product of a number (10) and a variable (x).

Worked Example

Step-by-Step

QUESTION: Is 5x^2y a monomial? Let's check.

STEP 1: Identify the number of terms. Here, we have '5' multiplied by 'x^2' multiplied by 'y'. It's all connected by multiplication.
---STEP 2: Check if there are any addition or subtraction signs separating parts of the expression. In '5x^2y', there are no '+' or '-' signs.
---STEP 3: Check the exponents of the variables. The exponent of 'x' is 2 (a non-negative integer), and the exponent of 'y' is 1 (also a non-negative integer).
---STEP 4: Since it has only one term and the variable exponents are non-negative integers, it fits the definition of a monomial.

ANSWER: Yes, 5x^2y is a monomial.

Why It Matters

Understanding monomials is crucial for building complex equations used in AI/ML algorithms to predict outcomes, in Physics to describe motion, and in Engineering to design structures. Engineers use these basic algebraic building blocks to model everything from bridge stability to rocket trajectories.

Common Mistakes

MISTAKE: Thinking expressions with addition/subtraction are monomials (e.g., 2x + 3y) | CORRECTION: Monomials have only one term, meaning no addition or subtraction signs separate different parts. 2x + 3y is a binomial.

MISTAKE: Including variables with negative exponents (e.g., 4x^-2) or fractional exponents (e.g., 7x^(1/2)) as part of a monomial | CORRECTION: For an expression to be a monomial, all variable exponents must be non-negative integers (0, 1, 2, 3...).

MISTAKE: Confusing a monomial with a polynomial | CORRECTION: A monomial is a specific type of polynomial – it's a polynomial with exactly one term. All monomials are polynomials, but not all polynomials are monomials.

Practice Questions

Try It Yourself

QUESTION: Is '7pqr' a monomial? | ANSWER: Yes, because it has one term and variable exponents are non-negative integers (p^1, q^1, r^1).

QUESTION: Identify which of these is NOT a monomial: (A) 12, (B) -3x, (C) x/y, (D) 6a^2b^3 | ANSWER: (C) x/y. This can be written as x * y^-1, which has a negative exponent for 'y', violating the monomial rule.

QUESTION: A mobile recharge pack gives 'R' GB of data per day for 'D' days. The total data is R multiplied by D. Is the expression for total data a monomial? Explain. | ANSWER: Yes. The total data is 'RD'. This is a product of two variables, forming a single term. Both R and D have an exponent of 1 (a non-negative integer).

MCQ

Quick Quiz

Which of the following is a monomial?

x + 5

3y^2

a/b

p^2 - q^2

The Correct Answer Is:

Option B, '3y^2', is a single term where the variable 'y' has a non-negative integer exponent (2). Options A and D have multiple terms separated by '+' or '-', and Option C (a/b or a*b^-1) has a variable with a negative exponent.

Real World Connection

In the Real World

When you use a fitness tracker or app to count your steps, the total distance covered might be calculated using simple formulas. If each step covers 's' centimeters, and you take 'N' steps, the total distance is 'Ns'. This 'Ns' is a monomial, a basic building block for calculating various metrics in health and sports analytics.

Key Vocabulary

Key Terms

TERM: Variable: A symbol (like x, y, a) that represents an unknown value. | TERM: Constant: A fixed numerical value (like 5, -10, 1/2). | TERM: Exponent: A number indicating how many times a base number or variable is multiplied by itself (e.g., 2 in x^2). | TERM: Term: A single number, variable, or product of numbers and variables.

What's Next

What to Learn Next

Great job learning about monomials! Now that you understand single-term expressions, you're ready to explore 'Polynomials'. Polynomials are expressions with one or more terms, and they are built using the monomials you just mastered.