How to Simplify and Solve Algebraic Expressions in Maths
Expressions and Equations:
Mathematics has been divided into several branches. The division that governs numbers and their processes is arithmetic. It is used for numerical calculation like addition, subtraction, multiplication and division. Geometry is all about the study of shapes and sizes of figures and their construction using a compass ruler and pencil. Algebra is another fascinating division in the shape of numbers and letters called as variables we describe our everyday situations.
Algebraic expressions consist of numbers and variables along with operational signs. Examples - addition, subtraction, multiplication, exponentiation with the natural exponent and division. Division with variables in an algebraic expression is called fractional expression.
Numbers which can be written in the fractional form are known as rational numbers. Examples- Terminating decimals, integers, and repeating decimals.
If an algebraic expression contains the root of the variable or the fractional power with a variable base then the expression is called irrational expression.
So, it may be rational and irrational to use algebraic terminology. Rational expressions are divided into integral and fractional expressions.
We need to understand what expression and equation are to grasp mathematics.
Definition of Expression:
An expression is a mathematical statement consisting of variables, numbers and an arithmetic operation between them. For example, (4m+ 5) is an expression where the terms 4 and 5 are constant and term m is a variable in the given expression, separated by the arithmetic operation + (plus).
A variable has no fixed value. In general, expression variables are represented by letters such as a, b, c, m, n, p, x, y, z, etc. We can construct a variety of expressions by combining various variables and numbers.
How to Simplify Algebraic Expressions?
The goal to simplify the algebraic expression is to find the simplified term of the expression given. To simplify algebraic expression, we should first know how to combine the same terms, how to factor a number, order of operations, to factor the expression or simplify it. The variables with the same degree are collected in conjunction with the same terms and the constant terms are isolated for the simplification process.
What is an Algebraic Expression?
An algebraic expression in mathematics is an expression consisting of variables and constants together with algebraic operations such as addition, subtraction, etc.
Examples of Algebraic Expressions Are:
\[3x + 4y - 7\], \[4x - 10\] etc
It should be remembered that, unlike the algebraic equations, there are no sides or equal signs of an algebraic expression. Some of the statements are:
\[3x + 4y - 7\]
\[4x - 10\]
\[2{x^2} - 3xy + 5\]
The Terminology Used In Algebraic Expressions:
An expression consisting of operation symbols, variables, and numbers is called an algebraic expression. In Algebra we deal with parameters, symbols or letters, the meaning of which we do not know.
Algebraic Expression:
In the above expression (i.e. \[5x - 3\]), x is a function whose value we do not know will take any value.
The x coefficient 5 is known as the variable term constant value and is well defined.
Is the expression of a constant value with a definite value?
The entire term is known as the Binomial term because it has two unlikely meanings.
Types of Algebraic expression:
There are three main types of algebraic terms, including:
Monomial Expression.
Binomial Expression.
Trinomial Expression.
Polynomial Expression.
Monomial Expression:
A monomial is classified as an algebraic expression that has only one term.
Monomial expression examples include:
Examples:
10ab2 is a two variables monomial in a and b.
5m2n is a two variables monomial in m and n.
-7pq is a two variables monomial in p and q.
5b3c is a two variables monomial in b and c.
2b is a one-variable monomial in b.
\[\frac{{2ax}}{{3y}}\]is a three variables monomial in a, x and y.
k2 is a one-variable monomial in k.
Binomial Expression:
A binomial expression is an algebraic expression that has two improbable terms.
Examples of binomial include:
m+n is two variables binomial in m and n.
a2+2bis two variables binomial in a and b.
5x3-9y3is two variables binomial in x and y.
-11p-q2 is two variables binomial in p and q.
\[\frac{{{b^3}}}{2} + \frac{c}{3}\] is two variables in b and c.
\[5{m^2}{n^2} + \frac{1}{7}\] is a two variables binomial in m and n.
Trinomial Expression:
An algebraic expression of only three non-zero terms is called a trinomial.
Examples of trinomial include:
x + y + z is three variables trinomial in x, y and z.
2a2 + 5a + 7 is a one-variable trinomial in a.
xy + x + 2y2 is two variables trinomial in x and y.
-7m5 + n3 - 3m2n2 is a two variables trinomial in m and n.
5abc-7ab+9ac is three variables trinomial in a, b and c.
\[\frac{{{x^2}}}{3} + ay - 6bz\] is five variables trinomial in a, b, x, y and z.
Polynomial Expression:
In general, a word with a variable's non-negative integral exponents is defined as a polynomial.
Examples of polynomial expression include:
2a+5b is a two terms polynomial in two variables a and b.
3xy + 5x + 1 is a three terms polynomial in two variables x and y.
\[3{y^4} + 2{y^3} + 7{y^2} - 9y + \frac{3}{5}\] is a five terms polynomial in two variables x and y.
m+5mn-7m2n+nm2+9is a four term polynomial in two variables m and n.
3+7x5+4x2 is a three terms polynomial in one variable x.
3+5x2-4x2y+5xy2 is a three terms polynomial in two variables x and y.
x-5yz-7z+11is a four term polynomial in three variables x, y and z.
1+2p+3p2+4p3+5p4+6p5+7p6 is a seven terms polynomial in one variable p.
Other Types of Expression:
Algebraic expression can also be divided into two different forms apart from monomial, binomial and polynomial expressions:
Numeric Expression.
Variable Expression.
Numeric Expression:
Numbers and operations consist of a numerical expression but never include any element.
Some of the numerical expression’s examples are as follows:
10+5
\[\frac{{15}}{2}\]
Variable Expression:
A variable expression is an expression that uses variables to describe an expression along with numbers and operation.
Some variable expression examples include:
4x+y
5ab+33
FAQs on Algebraic Expressions Explained: Definitions & Applications
1. What is an algebraic expression as per the Class 8 syllabus?
An algebraic expression is a mathematical phrase that combines variables (like x, y), constants (like 5, -9), and arithmetic operations (+, -, ×, ÷). Unlike an equation, it does not have an equals sign. For example, 7x - 4 and 2a² + 3b are algebraic expressions.
2. What are the different parts that make up an algebraic expression?
An algebraic expression is composed of the following parts:
- Terms: The individual building blocks of the expression, separated by addition or subtraction signs. For example, in `8x - 3y + 6`, the terms are `8x`, `-3y`, and `6`.
- Variables: The letters representing unknown values (e.g., `x` and `y`).
- Constants: The terms without any variables, which have a fixed value (e.g., `6`).
- Coefficients: The numerical part of a term that is multiplied by the variable. In the term `8x`, the coefficient is `8`.
3. How are algebraic expressions classified based on the number of terms?
Algebraic expressions are classified into different types based on the number of non-zero terms they contain:
- Monomial: An expression with a single term (e.g., `5x²`).
- Binomial: An expression with two unlike terms (e.g., `3a + 4b`).
- Trinomial: An expression with three unlike terms (e.g., `x² + 2x + 1`).
- Polynomial: The general term for an expression with one or more terms, where the exponents of the variables are non-negative integers.
4. What is the difference between like and unlike terms in an expression?
The primary difference is in their variables and corresponding exponents. Like terms have the exact same variables raised to the exact same powers, which allows them to be added or subtracted. For example, `7ab²` and `-2ab²` are like terms. On the other hand, unlike terms have different variables or different powers for the same variables, such as `7ab²` and `7a²b`, and cannot be combined into a single term.
5. What are the standard algebraic identities important for Class 8?
The standard algebraic identities are pre-defined formulas that are true for any value of the variables involved. For the CBSE Class 8 curriculum (2025-26), the key identities are:
- (a + b)² = a² + 2ab + b²
- (a – b)² = a² – 2ab + b²
- (a + b)(a – b) = a² – b²
- (x + a)(x + b) = x² + (a + b)x + ab
6. Why is it necessary to identify like terms before adding or subtracting expressions?
It is crucial to identify and group like terms because addition and subtraction are essentially operations of counting or combining similar objects. Think of variables as object types: you can add '3 apples' and '5 apples' to get '8 apples' (like `3x + 5x = 8x`), but you cannot combine '3 apples' and '5 oranges' into a single type (like `3x + 5y`). Ignoring this rule would be like mixing different units, leading to a mathematically incorrect result.
7. How are algebraic expressions used in real-life situations?
Algebraic expressions are used constantly in the real world to model situations with unknown or changing values. For example, if you are calculating a mobile phone bill, the total cost might be a fixed rental of ₹200 plus ₹1 for every GB of data used. This can be modelled by the expression `1g + 200`, where 'g' is the number of GBs used. This allows you to create a formula to calculate the bill for any amount of data usage.
8. What is the fundamental difference between an algebraic identity and an equation?
The fundamental difference lies in their scope of truth. An algebraic equation is a statement of equality that is true only for specific, limited values of its variables. For instance, `2x + 4 = 10` is only true when `x = 3`. In contrast, an algebraic identity is a statement of equality that holds true for all possible values of its variables. For example, `(a-b)² = a² - 2ab + b²` is valid for any numbers you choose for 'a' and 'b'.
9. How does multiplying two binomials, such as (x + 5)(x + 2), relate to finding the area of a shape?
Multiplying two binomials can be visualised as finding the area of a rectangle. Imagine a rectangle with its length defined by (x + 5) and its width by (x + 2). The total area (Length × Width) can be found by summing the areas of four smaller internal rectangles:
- A square of area x × x = x²
- A rectangle of area x × 2 = 2x
- Another rectangle of area 5 × x = 5x
- A small rectangle of area 5 × 2 = 10
