What Is a Monomial in Maths Definition Formula Properties and Solved Examples
The concept of Monomial in Maths plays a key role in algebra and is widely applicable in both real-life and exam situations. Learning about monomials helps build a strong foundation for more advanced maths topics, making problem-solving easier and more efficient.
What Is Monomial in Maths?
A monomial in maths is an algebraic expression that contains only one non-zero term. It can be only a number (constant), a variable, or the product of a constant and one or more variables, where the exponents of all variables must be non-negative integers. You’ll find this concept applied in areas such as algebraic expressions, polynomials, and simplifying equations.
Key Formula for Monomial in Maths
Here’s the standard form of a monomial: \( a \cdot x^m \cdot y^n \ldots \), where a is a real number (coefficient), variables have whole number (non-negative integer) exponents, and there’s just one term.
Monomial in Maths – Definition & Examples
A monomial in maths is an expression with a single term, made up of a number, a variable, or both multiplied together. No addition or subtraction signs appear within a monomial. For example:
- 5x
- 7
- -3xy²
- 4a³
- 2mnp
How to Identify a Monomial: Rules & Quick Tips
- It should have only one non-zero term.
- All variable exponents must be non-negative integers (e.g., x², not x−2 or x½).
- No variables should be present in the denominator.
- No addition or subtraction within the term.
Tip: A solitary number or a single variable is also a monomial (like 9 or y).
Monomial vs Binomial vs Trinomial
| Type | Definition | Example |
|---|---|---|
| Monomial | One term | 7x² |
| Binomial | Two terms (separated by + or -) | x + 5y |
| Trinomial | Three terms | 4x² + 2y + 6z |
Step-by-Step Illustration: Example Problem
Let's check if 8xy² is a monomial and find its degree:
1. Check for a single term.2. Confirm exponents: x (power 1), y (power 2) → all are non-negative integers.
3. No addition, subtraction, or variable in denominator.
4. It is a monomial.
5. Degree = 1 (x) + 2 (y) = 3.
Final Answer: 8xy² is a monomial of degree 3.
Try These Yourself
- Write any 5 examples of monomials.
- Find the degree of the monomial 3a4b2.
- Is 2x-1 a monomial?
- Identify the coefficient, variables, and degree in 6xy3.
Frequent Errors and Misunderstandings
- Thinking an expression with division by a variable (like 5/x) is a monomial—it is not.
- Mistaking negative exponents or fractional exponents as monomials.
- Assuming that having more than one variable automatically makes it a binomial (it still counts as one term if there's no + or -).
Degree of a Monomial: Simple Steps
The degree of a monomial is the sum of the powers of all its variables. If there’s no variable, degree is 0.
Example: In 4x²y³, degree = 2 + 3 = 5.
Tip: Look for the sum of exponents ONLY, don’t include the coefficient.
Monomial Operations (Addition, Multiplication)
You can add monomials with the same variables and exponents. To multiply monomials, multiply coefficients and add the exponents of matching variables.
Multiplication Example: Multiply 3xy × 5x².
1. Multiply coefficients: 3 × 5 = 152. Multiply variables: x × x² = x1+2 = x³; y remains y.
3. Final answer: 15x³y.
Relation to Other Concepts
The idea of monomial in maths connects closely with Algebraic Expressions and Polynomials. Knowing how to identify and work with monomials helps with solving equations, factorizing, and understanding topics like the degree of a polynomial.
Cross-Disciplinary Usage
Monomials in maths help in Physics for writing and simplifying formulas involving powers of variables, in Computer Science for polynomial time complexity, and in day-to-day tasks involving simple repeated multiplication. This is useful in exams like CBSE, JEE, and Olympiads.
Classroom Tip
A quick mnemonic: “Mono” means one—so a monomial always has just one term. In Vedantu’s live sessions, teachers often say, “If you only see multiplication (and no plus or minus in-between), you’re probably looking at a monomial!”
Wrapping It All Up
We explored monomial in maths from its definition, formula, classic examples, and common mistakes, to its links with expressions and polynomials. For more concept clarity on monomials, continue practicing on Vedantu and try related topics below to supercharge your exam scores.
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FAQs on Monomial in Maths Complete Guide to Definition and Examples
1. What is a monomial in maths?
A monomial is an algebraic expression that consists of only one term made up of a constant, a variable, or a product of constants and variables with non-negative integer exponents.
- Examples: 5x, 7, 3a², -4xy
- It has no addition or subtraction separating terms.
- The exponent of each variable must be a whole number (0, 1, 2, ...).
2. What are the parts of a monomial?
The parts of a monomial are the coefficient, variables, and exponents.
- Coefficient: The numerical factor (e.g., 6 in 6x²).
- Variable: The letter representing an unknown (e.g., x in 6x²).
- Exponent: The power of the variable (e.g., 2 in x²).
3. What is the degree of a monomial?
The degree of a monomial is the sum of the exponents of all its variables.
- For 5x³, degree = 3.
- For 4x²y³, degree = 2 + 3 = 5.
- A non-zero constant like 7 has degree 0.
4. Can you give some examples of monomials?
Examples of monomials include expressions with only one term and non-negative integer exponents.
- 8 (constant monomial)
- −2x
- 3a²
- 5xy³
5. How do you multiply monomials?
To multiply monomials, multiply the coefficients and add the exponents of like variables.
- Step 1: Multiply coefficients.
- Step 2: Add exponents of the same variables using the law aᵐ × aⁿ = aᵐ⁺ⁿ.
6. How do you divide monomials?
To divide monomials, divide the coefficients and subtract the exponents of like variables.
- Step 1: Divide numerical coefficients.
- Step 2: Apply the rule aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
7. What is the difference between a monomial, binomial, and polynomial?
The difference lies in the number of terms in the algebraic expression.
- Monomial: One term (e.g., 5x).
- Binomial: Two terms (e.g., x + 3).
- Polynomial: One or more terms (e.g., x² + 3x + 1).
8. Is a constant a monomial?
Yes, a constant is a monomial because it is a single term with no variables.
- Examples: 5, −9, 100.
- The degree of any non-zero constant is 0.
9. Can a monomial have a negative or fractional exponent?
No, a monomial cannot have negative or fractional exponents because exponents must be non-negative integers.
- x⁻² and x¹ᐟ² are not monomials.
- Valid exponents are 0, 1, 2, 3, and so on.
10. How do you identify a monomial?
You can identify a monomial by checking if the expression has only one term and non-negative integer exponents.
- No addition or subtraction between terms.
- Variables have whole number exponents.
- It may include constants and multiple variables multiplied together.
