If you’re learning algebra, you’ve probably heard the term “ expression ” before.
But what exactly is an expression in algebra?
Simply put, an expression is a combination of numbers, variables, and mathematical operations that can be simplified or evaluated.
Expressions are used in algebra to represent realworld situations and solve problems.
Expressions can take many forms, from simple equations like 5x + 2 to more complex expressions like \frac{3x + 2y}{x  y} .
They can include variables, which are letters used to represent unknown values, as well as constants, which are fixed values like numbers.
Expressions can also include mathematical operations like addition + , subtraction  , multiplication \times , and division \div , as well as exponents x^{\fcolorbox{lightgray}{lightgray}{2}} and parentheses \fcolorbox{lightgray}{lightgray}{(}x \times 3y\fcolorbox{lightgray}{lightgray}{)} + 5 .
Expressions are an important part of algebra, as they allow us to represent and solve problems using mathematical language. By understanding what expressions are and how they work, you’ll be able to tackle more complex algebraic problems and gain a deeper understanding of the subject as a whole.
What is an Expression in Algebra?
In algebra, an expression is a mathematical phrase that combines numbers and/or variables using mathematical operations such as addition, subtraction, multiplication, and division. Expressions represent a value or a quantity that can vary based on the values assigned to the variables.
For example, the expression 3x + 5 represents a value that depends on the value of x . If x is equal to 2 , then the value of the expression is 11 evaluated by firstly substituting the value 2 for x , therefore, 3\fcolorbox{yellow}{yellow}{(2)} + 5 meaning that 3 \times 2 = 6 and then 6 + 5 = 11 .
If x was equal to 4 , then the value of the expression is 17 from the calculations 3\fcolorbox{yellow}{yellow}{(4)} + 5 , therefore, 3 \times 4 = 12 , then 12 + 5 = 17 .
Expressions can be made up of one or more terms . A term is a combination of a number and a variable or variables that are multiplied together.
For example, in the expression 3x + 5 , the terms are 3x and 5 .
It’s important to note that expressions do not have an equal sign , unlike equations.
Equations, on the other hand, are mathematical statements that assert the equality of two expressions.
Types of Algebraic Expressions
Algebraic expressions can be classified into three main types: monomial expressions, binomial expressions, and polynomial expressions. Each type has its own unique characteristics and properties.
Monomial Expressions
A monomial expression is an algebraic expression that contains only one term. The term can be a constant, a variable, or a product of constants and variables. Monomial expressions are also known as singleterm expressions.
Examples of monomial expressions include:
 5
 x
 3xy
 2a^{2}b
Binomial Expressions
A binomial expression is an algebraic expression that contains two terms. The terms can be constants, variables, or products of constants and variables. Binomial expressions are also known as twoterm expressions.
Examples of binomial expressions include:
 x + 2
 3y  4
 2ab + 3bc
 5a^{2} + 7b
Polynomial Expressions
A polynomial expression is an algebraic expression that contains two or more terms. The terms can be constants, variables, or products of constants and variables. Polynomial expressions are also known as multiterm expressions.
Examples of polynomial expressions include:
 x^{2} + 2x + 1
 4y^{3}  2y^{2} + 5y  1
 2a^{2}b + 3ab^{2}  4abc
 3x^{4} + 2x^{3}  5x^{2} + 7x  1
Parts of an Algebraic Expression
When working with algebraic expressions, it is important to understand the different parts that make up an expression. These parts include coefficients, variables, exponents, and constants. By understanding these parts, you can more easily manipulate and simplify algebraic expressions.
Coefficients
The coefficient is the number that appears in front of a variable. For example, in the expression 3x , the coefficient is 3 . In the expression 2y , the coefficient is 2 . When there is no number written in front of a variable, the coefficient is assumed to be 1 .
For example, in the expression x , the coefficient is 1 .
Variables
Variables are letters that represent numbers in an algebraic expression. The most common variables used in algebra are x and y , but any letter or symbol can be used. Variables can be raised to a power or exponent, which indicates how many times the variable is multiplied by itself.
For example, in the expression 2x^{3} , the variable x is raised to the power of 3 .
Exponents
An exponent is a number that indicates how many times a variable is multiplied by itself. Exponents are written as small numbers to the right of a variable.
For example, in the expression x^{2} , the exponent is 2 , which means that x is multiplied by itself 2 times.
Constants
A constant is a number that appears in an algebraic expression without a variable .
For example, in the expression 2x + 5 , the constant is 5 .
Constants can be positive or negative and can be added or subtracted from other terms in an expression.
Simplifying Algebraic Expressions
When you have an algebraic expression, you may want to simplify it to make it easier to understand and work with. Simplifying an expression means writing it in a simpler form without changing the value of the expression. This is done by combining like terms and applying the distributive property.
To simplify an expression, you need to follow the order of operations , which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
Here are some steps to follow when simplifying algebraic expressions:

Combine like terms: Terms that have the same variable and exponent can be combined by adding or subtracting their coefficients.
 For example, in the expression 3x + 2x  5 , the like terms are 3x + 2x , which can be combined to give 5x . The expression then becomes 5x  5 .

Apply the distributive property: When a number is multiplied by a sum or difference, you can distribute the number to each term in the sum or difference.
 For example, in the expression 2(x + 3) , the 2 can be distributed to both x and 3 to give 2x + 6 .

Combine like terms again: If there are still like terms in the expression, you can combine them again to simplify further.
 For example, in the expression 4x + 2y  2x  y , the like terms are 4x and 2x , which can be combined to give 4x  2x = 2x , and the like terms are 2y and y , which can be combined to give 2y  y = y . The expression then becomes 2x + y .
It’s important to remember that when simplifying an expression, you are only changing the way it is written, not its value .
So, if you simplify an expression and get a different expression, you need to check that they have the same value .
Testing Your Simplified Or Expanded Expression
When you simply or expand an expression it can help by inserting values into the variables of your simplified or expanded expression to see if you’ve done it correctly.
For example, suppose you had the following expression x^{2} + 2x + 1 and you thought you could simplify the expression by adding the x^{2} term and the 2x together and you arrived at what you thought was a simplified expression of 2x^{2} + 1 .
To check your answer, you could substitute the number 3 for x in both the original and your simplified answer.
Here’s how the original expression would look when 3 is substituted:
x^2 + 2x + 1 \\ (3)^2 + 2(3) + 1 \\ 9 + 6 + 1 \\ 16
As you can see, the original expression simplifies to the value of 16 when 3 is substituted for the value of x .
Here’s how your simplified expression would look when the same value of 3 is substituted in:
2x^{2} + 1 \\ 2(3)^{2} + 1 \\ 2 \times 9 + 1 \\ 18 + 1 \\ 19
As you can see from the result above, your answer above doesn’t match the original expression. Therefore, your simplified expression is incorrect .
You can read more about combining like terms to avoid making this type of mistake.
Using Algebraic Expressions
In algebra, expressions are used to represent mathematical relationships between numbers and variables. They are made up of constants, variables, and mathematical operations such as addition, subtraction, multiplication, and division.
Expressions can be used to simplify complex problems and make them easier to solve.
One common use of algebraic expressions is to solve word problems.
For example, if you were given a word problem that asked you to find the area of a rectangle with a length of 5 units and a width of 3 units, you could use the expression A = lw , where A represents the area, l represents the length, and w represents the width.
By plugging in the values for l and w , you can simplify the expression to find the area and if you measured the length of a rectangle to have 5 units and the width of the same rectangle to be 3 units, therefore, substituted into the area formula would be:
A = 5 \times 3 \\ 15
Another use of algebraic expressions is to represent realworld situations.
For instance, if you were given a problem that asked you to find the cost of a certain number of items that cost $2 each, you could use the expression C = 2n , where C represents the cost and n represents the number of items.
By plugging in the value for n , you can simplify the expression to find the cost:
C = 2 \times 10 \\ 20
Algebraic expressions can also be used to create formulas, which are equations that describe relationships between variables.
For example, the formula for the area of a circle is A = \pi r^{2} , where A represents the area and r represents the radius. By plugging in the value for r , you can simplify the expression to find the area:
A = \pi 5^{2} \\ 78.54
Overall, algebraic expressions are a powerful tool for solving mathematical problems and representing realworld situations. By understanding how to use them, you can simplify complex problems and find solutions more easily.