Chapters

What is a Monomial?
Rules of Operations
Practice Questions & Solutions
Glossary of Key Terms
Curriculum & Resources

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What is a Monomial?

A monomial is an algebraic expression consisting of exactly one term. It can be a number, a variable, or a product of numbers and variables.

The Anatomy of a Monomial

Coefficient: The numerical part of the term (e.g. in 5x, the coefficient is 5).
Variable: The letter representing an unknown value (e.g. x or y).
Exponent: The power to which a variable is raised. Crucial Rule: The exponent of a monomial must be a whole number (0, 1, 2, 3...).

Examples vs. Non-Examples

Is it a Monomial?	Term	Why or Why Not?
Yes	$\text{[math]}$	Constants are monomials with degree 0.
Yes	$\text{[math]}$	Numbers and variables multiplied together.
No	$\text{[math]}$	This is a binomial (two terms).
No	$\text{[math]}$	Exponents must be positive whole numbers.
No	$\text{[math]}$	Dividing by a variable is not allowed in monomials.

Rules of Operations

To manipulate monomials effectively, you must follow specific algebraic rules. These rules are standard across the GCSE and A-Level curriculum for algebraic manipulation.

1. Addition and Subtraction

You can only add or subtract like terms (terms with the exact same variables and exponents). When you do this, you simply add or subtract the coefficients.

Worked Example

Simplify:

7x^{2}y + 3x^{2}y - 2x^{2}y

7 + 3 - 2 = 8

Result:

8x^{2}y

2. Multiplication of Monomials

When multiplying monomials, you multiply the coefficients and add the exponents of the same variables (Product Rule of Exponents).

Worked Example

5x^{2}\times 2x^{4}

Multiply coefficients:

5\times2=10

Add exponents:

2+4=6

Result:

10x^6

3. Division of Monomials

When dividing, you divide the coefficients and subtract the exponents of the same variables (Quotient Rule of Exponents).

Worked Example

\dfrac{15x^{6}} {3x^{2}}

Divide coefficients:

\dfrac{15}{3}=5

Subtract exponents:

6-2=4

Result:

5x^4

4. Power of a Monomial

To raise a monomial to a power, raise the coefficient to that power and multiply the exponents of the variables (Power of a Power Rule).

Worked Example

(2x^3)^4

Raise coefficient:

2^4=16

Multiply exponents:

3\times4=12

Result:

16x^{12}

Practice Questions & Solutions

Score:

Simplify the following expression by combining like terms:

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Solution

Identify and group the like terms (those with the same literal parts):

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Find the product of the following two monomials:

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Solution

Multiply the coefficients first, then add the exponents of the same variables:

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Simplify the algebraic fraction:

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Solution

Divide the coefficients and subtract the exponents of the same variables:

$\text{[math]}$

Simplify the expression:

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Solution

Raise the coefficient to the power and multiply each variable's exponent by the power:

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Simplify the following:

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Solution

First, simplify the numerator using the multiplication rule:

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Now, divide by the denominator:

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Glossary of Key Terms

Degree of a Monomial: The sum of the exponents of all included variables. For example, the below has a degree of 2 + 3 = 5:

x^{2}y^{3}

Literal Part: The variable and exponent portion of the monomial (everything except the coefficient).
Polynomial: An expression consisting of one or more monomials joined by addition or subtraction.
Constant: A monomial with no variables (e.g.,7).

Curriculum & Resources

This topic is a cornerstone of the GCSE Mathematics (Number and Algebra) and Common Core (High School Algebra) objectives for rewriting and performing operations on polynomials.

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Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

Operations with Monomials