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Algebraic expressions are mathematical statements formed by combining variables and constants using operations such as addition, subtraction, multiplication, and division. For example, consider the expression 5x+7; this is a typical algebraic expression combining a variable term
In this article, we’ll learn more about algebraic expressions in detail, their types and the operations that can be performed on the Algebraic Expressions.
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What are Algebraic Expressions?
An algebraic expression is also known as a variable expression. It is a combination of terms created using operations such as addition, subtraction, multiplication, and division. These expressions consist of variables, constants, and coefficients connected by arithmetic operations.
Algebraic Expressions Examples
Imagine John and Naira playing with matchsticks to create number patterns. John used four matchsticks to form the number 4. Naira then added three more matchsticks, creating a pattern with two 4’s. They noticed that by adding 3 matchsticks each time, they could form an additional “four.”
This led them to a general rule: to create a pattern with n number of 4’s, they would need 4 + 3 (n − 1) matchsticks. This expression, 4 + 3 (n − 1), is an example of an algebraic expression.
Below are some more examples of algebraic expressions:
5x + 4z = 16
2a – 8b = 25
4x 5t = 9t
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Variables, Constants, Terms, and Coefficients
Algebraic expressions are made up of various components that define their structure and function. To work with expressions effectively, we need to understand the concepts of Variables, Constants, Terms, and Coefficients effectively.
Variables: Variables are symbols (like x,y, or a) that represent unknown values and can change within the context of the expression. In simpler words, a variable is a symbol that does not have a fixed value and can represent different numbers. Variables can take any value within a given context. For example, in the matchstick pattern problem, n is a variable that can take values like 1, 2, 3, and so on.
Constants: A constant is a symbol with a fixed numerical value. Unlike variables, constants do not change. All numbers are constants. In simpler words, Constants are fixed numerical values within the expression that do not change, such as numbers like 3, 7, 11, etc.
Terms: Terms are the individual parts of an expression separated by addition or subtraction signs. A term can be a single variable, a single constant, or a combination of variables and constants connected by multiplication or division. For example, in 5x + 4y + 10, the terms are 5x, 4y, and 10 connected by +.
Coefficients: Coefficients are numerical factors that multiply the variables within a term. In the term 5x, the number 5 is the coefficient.
These components work together to form the building blocks of algebraic expressions. It makes it possible to perform operations and solve equations in mathematics.
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Simplifying Algebraic Expressions
To simplify an algebraic expression, we combine like terms. Like terms are the terms that have the same variable raised to the same power. This process reduces the expression to its simplest form. Let’s look at an example to understand this concept:
Let the given expression be:
x2 + 4x3 + 8x + 5x2 − 2x3 − x
Now, we will simplify it by grouping and combining like terms. Therefore, we get:
(x2 + 5x2) + (4x3 − 2x3) + (8x − x)
Therefore, the simplified form of
x2 + 4x3 + 8x + 5x2 − 2x3 − x is:
(x2 + 5x2) + (4x3 − 2x3) + (8x − x).
Adding Algebraic Expressions
Adding algebraic expressions involves combining like terms from each expression. Below are a few examples of applying addition operation on the given algebraic expression.
- (x2 + 2x3) + (5x2 + 4x)Now, combining the like terms we get: (x2 + 5x2) + (4x + 2x3)Solving it, we get: 6x2 + 6x3
- (5x2 + 2x3) + (x2 + 2x)Now, combining the like terms we get: (5x2 + x2) + (2x + 2x3)Solving it, we get: 6x2 + 4x3
Subtracting Algebraic Expressions
To subtract one algebraic expression from another, add the additive inverse (change the signs) of the second expression to the first. Here are some examples:
- (x2 + 2x3) – (5x2 + 4x)Now, combining the like terms we get: (x2 – 5x2) + (2x3 – 4x)Solving it, we get: -4x2 – 2x3
- (5x2 + 2x3) – (3x2 + x3)Now, combining the like terms we get: (5x2 – 3x2) + (2x3 – x3)Solving it, we get: 2x2 + x3
- (3ab + 4) + (-2ab + 4)Now, combining the like terms we get: (3ab – 2ab) + (4 + 4)Solving it, we get: ab + 8
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Multiplying Algebraic Expressions
To multiply two algebraic expressions, multiply every term in the first expression by every term in the second expression, then combine all the resulting products. Below given are some examples:
- 2x ( 8y + z) = 16xy + 2xz
- a ( 8b – z) = 8ab – az
Dividing Algebraic Expressions
To divide algebraic expressions, factor both the numerator and the denominator, cancel any common factors, and simplify the expression. Below are some examples:
- 16x2 / 2x = 8x
- 50a3b2 / 5b = 10ab
Algebraic Expression Formulas
Algebraic formulas are simplified expressions that help solve equations quickly by providing a standard approach. Below is the list of some commonly used algebraic formulas:
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- (a + b)(a – b) = a2 – b2
- (x + a)(x + b) = x2 + x(a + b) + ab
- (a + b)3 = a3 + 3a2b + 3ab2 + b3
- (a – b)3 = a3 – 3a2b + 3ab2 – b3
- a3 + b3 = (a + b)(a2 – ab + b2)
Types of Algebraic Expressions
Algebraic expressions can be categorised based on the variables they contain, the number of terms, and the exponents of the variables. Below is a table that defines the five main types of algebraic expressions with their examples:
| Type | Meaning | Examples |
|---|---|---|
| Monomial | An expression with only one term where the exponents of all variables are non-negative integers. | 3xy |
| Binomial | An expression consisting of exactly two monomials. | 3x − 2y2 |
| Trinomial | An expression made up of three monomials. | 3x − 2y + z |
| Polynomial | An expression containing one or more monomials. | −3x3 + 7x2 + 3x + 5 |
| Multinomial | An expression with one or more terms where the exponents of variables can be either positive or negative. | 4x−1 + 2y + 3z |
Algebraic Expressions: Solved Examples
Q. 1. Simplify: (a2 + 2b3) + (5a2 + 4b3)
Ans. The given algebraic expression is (a2 + 2b3) + (5a2 + 4b3)
Now, combining the like terms we get: (a2 + 5a2) + (4b3 + 2b3)
Solving it, we get: 6a2 + 6b3
Q. 2. Simplify: (3ab + 4) − (−2ab + 4)
Ans. The given algebraic expression is (3ab + 4) − (−2ab + 4)
Now, combining the like terms we get: (3ab + 2ab) + (4 − 4)
Solving it, we get: 5ab
Algebraic Expressions: Practice Questions
Q. 1. Simplify: (5a2 + 2b3) + (9a2 + b3)
Q. 2. Simplify: (a2 + 5) + (5a2 + 9b3)
Q. 3. Simplify: (7x2 – 2b3) + (5x2 + 2b3)
Algebraic Expressions: FAQs
What is an algebraic expression?
An algebraic expression is a mathematical equation that includes numbers, variables, and operations like addition, subtraction, multiplication, and division. Examples include 5a² + 2b³.
What are like terms in an algebraic expression?
Like terms are terms that have the same variable(s) raised to the same power. For example, in the expression 3x + 5x, both terms are like terms because they contain the variable x to the same power.
How do you simplify an algebraic expression?
To simplify an algebraic expression, combine like terms by adding or subtracting their coefficients. For example, to simplify 4x + 3 − 2x + 5, combine the like terms to get 2x + 8.
