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Algebraic Expression and its Operations with Examples

Introduction

As we all know, algebra is a branch of mathematics where we study variables and constants and learn operations on them, such as addition, subtraction, multiplication and division, the similar operations that are used in arithmetic. In algebra operations are performed on algebraic expressions whereas in arithmetic, operations are performed on only numbers.

Before getting a deep dive into algebraic operations, first we understand what are the parts of an algebraic expression, how it is written and know some of its basic terms.
So, it comprises of two parts variable and constant.

Variable

A symbol which can take different numerical value is called a variable. We use any letters such as a, b, c, x, y, z etc. to represent variable.

Constant

Constant which has a fixed numerical value.

Example

5, 25, 100 etc.

These constants and variables when combined with any of these operations + or – or × or ÷ result to an algebraic expression.

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Algebraic expression

A combination of constants and variables which are connected by some or all four fundamental operations +, -, ×, ÷ is called an algebraic expression.

Example

3xy + 2y z + 8x z + 9
This example is of an algebraic expression connected by + operation by combining variables 3xy, 2y z , 8x z and a constant 9.

Term

Term is a part in an algebraic expression that comes in between operations + and -.

In other words, the various parts of algebraic expression which are separated by + or – are called terms.

Example

3xy + 4yz – 8zx
3xy, 4yz and -8zx are terms
Moreover, we can say 3xy + 4yz – 8zx has total of three terms.

Factor

We have learnt above the term and how to find it in algebraic expressions.
So, here comes the factors. As we know terms are formed by the product of variables and constants, so these variables and constants in a term are known as factors.
So, a term can be written as product of factors.
These factors contain a numerical coefficient and an algebraic factor.

Example

In 3xy + 4yz – 8zx, 3 is numerical factor and xy is algebraic factor.
4 is numerical factor and yz is algebraic factor.
-8 is numerical factor and zx is algebraic factor.

Further, we can classify terms on the basis of factors into Like Terms and Unlike Terms. Let’s see how next.

Like Terms

When terms have same algebraic factor, then they are called as like terms.

Example

3xy + 4y + 2xy + 5z
Here, 3xy and 2xy are like terms as they have same algebraic factors x and y.

Unlike Terms

When terms have different algebraic factor, then they are called as unlike terms.

Example

3xy + 4y + 2xy + 5z
Here, 4y and 5z are unlike terms as they have different algebraic factors y in term 4y and z in term 5z.


Let’s see next how to do operations of addition, subtraction, multiplication and division on them.

Operations on algebraic expression


Operations on algebraic expressions are similar to the arithmetic operations like addition, subtraction, multiplication and division. Let’s take a look at how these operations are performed with examples.

Addition

How are two or more than two algebraic expressions added?
It is simple, just combine all the like terms and do the addition on the numerical coefficients of these like terms.
So, rule of thumb is combine only like terms and the usual addition operation on coefficients of all like terms.

Example

Example 1: Add
3xy + 7yz
8xy + 5yz

= (3xy + 7yz) + (8xy + 5yz)
Step 1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step 2: Add the like terms separately.
= (3xy + 8xy) + (7yz + 5yz)
= (3+8)xy + (7 + 5)yz
= 11xy + 12yz


Example 2: Add
3m2 + 5n2 + 11
7m2 – 2n2 + 6

= 3m2 + 5n2 + 11 + 7m2 – 2n2 + 6
Step 1: Find like terms in both expressions.
So, the like terms are:
3m2 and 7m2
5n2 and -2n2
Step 2: Add the like terms separately.
= (3m2 + 7m2) + (5n2 + (-2n2)) + (11 + 6)
= 10m2 + (5n2 – 2n2) + 17
= 10m2 + 3n2 + 17


Example 3: Add
8x2 + 11y2 + 6z2
9x2 – 2y2 – 7z2
2x2 + 4y2 + 5z2

= (8x2 + 11y2 + 6z2) + (9x2 – 2y2 – 7z2) + (2x2 + 4y2 + 5z2)
Combine like terms.
= (8x2 + 9x2 + 2x2) + (11y2 – 2y2 + 4y2) + (6z2 – 7z2 + 5z2)
= 19x2 + 13y2 + 4z2


Example 4: Add
20a + 12 and
7b + 8

= (20a + 12) + (7b + 8)
We usually combined like terms in the above examples, but we here unlike terms also exist.
Here, 20a and 7b are unlike terms.
= 20a + 7b + (12 + 8)
= 20a + 7b + 20


Subtraction

The subtraction operation is similar to the addition operation. To subtract algebraic expressions, combine all the like terms and do the subtraction on the numerical coefficients of these like terms.

Example

Example 1: Subtract
3xy + 7yz
8xy + 5yz

= (3xy + 7yz) – (8xy + 5yz)
Step 1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step 2: Subtract the like terms separately.
= (3xy – 8xy) + (7yz – 5yz)
= (3 – 8)xy + (7 – 5)yz
= -5xy + 2yz


Example 2: Subtract 3m2 + 5n2 + 11
7m2 – 2n2 + 6

= (3m2 + 5n2 + 11) – (7m2 – 2n2 + 6)
Step 1: Find like terms in both expressions.
So, the like terms are:
3m2 and 7m2
5n2 and -2n2
Step 2: Subtract the like terms separately.
= (3m2 – 7m2) + (5n2 – (-2n2)) + (11 – 6)
= -4m2 + (5n2 + 2n2) + 5
= -4m2 + 7n2 + 5


Example 3: Subtract
8x2 + 11y2 + 6z2
9x2 – 2y2 – 7z2

= (8x2 + 11y2 + 6z2) – (9x2 – 2y2 – 7z2)
Combine like terms.
= (8x2 – 9x2) + (11y2 – (-2y2)) + (6z2 – (-7z2))
= -x2 + (11y2 + 2y2) + (6z2 + 7z2)
= -x2 + 13y2 + 13z2


Example 4: Subtract 7b + 8 from 20a + 12
= (20a + 12) – (7b + 8)
here we have unlike terms.
20a and 7b are unlike terms.
= 20a – 7b + (12 – 8)
= 20a – 7b + 4


Multiplication

We can also multiply the algebraic expressions. Here we do not need to combine like terms as we did above in the addition and subtraction operations. In multiplication, all terms of first expression are multiplied by the terms of second expression.
Let’s learn it by following examples.

Example

Example 1: Multiply 3x + 2y by 4x
= (3x + 2y) × 4x
Multiply each term of (3x + 2y) to 4x
= (3x × 4x) + (2y × 4x)
= 12x2 + 8xy


Example 2: Multiply 4x2 + 5x + 7 by 8xy
= (4x2 + 5x + 7 × 8xy
Multiply each term of (4x2 + 5x + 7) to 8xy
= (4x2 × 8xy) + (5x × 8xy) + 7 × 8xy
= 32x3 + 40x2y + 56xy


Example 3: Multiply 7x – 3y by 2y + 5z
= (7x – 3y) × (2y + 5z)
Multiply each term of (7x – 3y) to (2y + 5z)
= 7x × (2y + 5z) – 3y × (2y + 5z)
Again, multiply each term of (2y + 5z) with 7x
and multiply each term of (2y + 5z) with 3y
= 7x × 2y + 7x × 5z – 3y × 2y – 3y × 5z
= 14xy + 35xz – 6y2 – 15yz


Example 4: Multiply 3a 5 + 7b 2 by 7a 2
= ( 3a 5 + 7b 2 ) × 7a 2
Multiply each term of ( 3a 5 + 7b 2 ) to 7a 2
= ( 3a 5 × 7a 2 ) + ( 7b 2 × 7a 2 )
= ( 3 5 × 7 2 a2) + ( 7 2 × 7 2 ab)
= 21a2 10 + 49ab 4


Division

Divisions involved in algebraic expressions are all similar to that we do in arithmetic. In arithmetic, we know that when 21 is divided by 3, we get number 7 which can be written as 21 3 = 7.
We can divide algebraic expressions in the similar way too.

Example

Example 1: Divide 12x by 4x
= 12x 4x
= 12 × x 4 × x
= 3


Example 2: Divide 24x2y2z2 by 4x2y
= 24x2y2z2 4x2y
= 24 × x × x × y × y × z × z 4 × x × x × y
= 6yz2


Example 3: Divide 3x4 + 6x3 – 9x2 + 12x by 3x
= 3x4 + 6x3 – 9x2 + 12x 3x
= 3x4 3x + 6x3 3x 9x2 3x + 12x 3x
= x3 + 2x2 – 3x + 4


Example 4: Divide x2 + 5x + 6 by x + 2
= x2 + 5x + 6 x + 2
= x2 + 2x + 3x + 6 x + 2     (Factorise)
= x(x + 2) + 3(x + 3) x + 2
= (x + 3)(x + 2) x + 2
= x + 3


Solved Examples

1) Add expressions 7a + b, 8a + b + c and 9a + 2c.

= 7a + b + 8a + b + c + 9a + 2c
First combine the like terms
= (7a + 8a + 9a) + (b + b) + (c + 2c)
= 24a + 2b + 3c

2) Solve the expression p2 + q2 + 2p2 - q2 - p2 + q2.

= p2 + q2 + 2p2 - q2 - p2 + q2
Combine the like terms
= (p2 + 2p2- p2) + (q2 - q2 + q2)
= 2p2 + q2

3) Subtract 6a + 7b + 2c from 18a + 9b + 4c.

= 18a + 9b + 4c - (6a + 7b + 2c)
= 18a + 9b + 4c - 6a - 7b - 2c
Combine the like terms
= (18a - 6a) + (9b - 7b) + (4c - 2c)
= 12a + 2b + 2c
= 2(6a + b + c)

4) Subtract the sum of 6a + b and 7a + 4b from 20a + 19b.

Sum of 6a + b and 7a + 4b = (6a + b) + (7a + 4b)
= (6a + 7a) + (b + 4b)
= 13a + 5b
Subtract 13a + 5b from 20a + 19b
= (20a + 19b) - (13a + 5b)
= 20a + 19b - 13a - 5b)
= (20a - 13a) + (19b - 5b)
= 7a + 14b
= 7(a + 2b)

5) Multiply 4x2 by (x + 3)

= 4x2(x + 3)
= 4x2 × x + 4x2 × 3
= 4x3 + 12x2

6) Multiply (x + 3) by (2x2 + 5)

= (x + 3) × (2x2 + 5)
= x × (2x2 + 5) + 3 × (2x2 + 5)
= x × 2x2 + x × 5 + 3 × 2x2 + 3 × 5
= 2x3 + 5x + 6x2 + 15
= 2x3 + 6x2 + 5x + 15

7) Divide 2x2 + 4x by 2x

(2x2 + 4x) ÷ 2x
= 2x2 + 4x 2x
= 2x(x + 2) 2x
= (x + 2)

8) Divide 4x + 12 by 2

(4x + 12) ÷ 2
= (4x + 12) 2
= 4(x + 3) 2
= 2(x + 3)

9) Divide 24x2y2z2 by -8x

(24x2y2z2) ÷ -8x
= - 24x2y2z2 8x
= - 24 × x × x × y × y × z × z 8x
= -3xy2z2

10) Divide x2 + 4x + 2 by x + 2

(x2 + 4x + 2) ÷ (x + 2)
= (x2 + 4x + 2) (x + 2)
= (x2 + 2x + 2x + 2) (x + 2)
= (x + 2)(x + 2) (x + 2)
= (x + 2)

Worksheet 1

Fill in the blanks

  1. The coefficient of x in -4xy is ___________.
  2. The numerical coefficient of 7x2y is ___________.
  3. The coefficient of x2y in 9x2y + 3z is ___________.
  4. The number of terms in the expression 3x2y + 8yz + 4z is ___________.
  5. 7x2 and ___________ are like terms in the expression 7x2 + 8z2 + 3x2.
  6. 4x + 5 contains ___________ terms.
  7. The numerical coefficient of -4xyz is ___________.
  8. The constant term in expression x2y + 9x2 + 4y2 + 10 is ___________.
  9. The sum of 4x, 2x and 6x is ___________.
  10. The coefficient of xy in expression x2 - 12xy + y2 is ___________.
Help icon Help box
-4
3x2
-4y
-12
two
12x
10
9
7
3

Worksheet 2

Addition of algebraic expressions.

  1. 4x + 2x + 6x
  2. 9x2 + (- 2x2) + 2x2
  3. x + y and x + y
  4. 2x + 15x
  5. (-2x) + (- 7x)
  6. (3m2 + 9n) + (6m2)
  7. 28xy + 8xy + 2xy
  8. (p + q) + (p + q)
  9. 25m + 6m2 + 16m2 + 15m
  10. 12p + 12q + 6p + 2q

Worksheet 3

Subtraction of algebraic expressions.

  1. (7a + 2ab) - (7a - 2b)
  2. 3ab + 6b - (14ab + 6b)
  3. (12b + 12c) - (12b + 12c)
  4. (3y2 + 5y) - (3y2 - 5y)
  5. -7yz + 8xz - 7yz -8xz
  6. Subtract 16xy from 30xy.
  7. Subtract (a - b) from 2a + 2b.
  8. Subtract 6m2 + n2 from 16m2 + 15n2.
  9. From the sum of 3x + y and 6x + 7y, subtract 15x + 15y.
  10. Subtract 21x2 - 12y2 from the sum of (2x2 + 4y2) and (13x2 + 10y2).

Worksheet 4

Multiplication of algebraic expressions.

  1. 9x × x
  2. x2 × y
  3. 6a2 × 2a
  4. 10x × 10y
  5. 100xy × 0
  6. 3a2b × a2b
  7. 5mn × 6np
  8. 3 4 x × 4 3 y
  9. 2q × 3q2 × 6q3
  10. m2n × n2p × p2m

Worksheet 5

Division of algebraic expressions.

  1. 6x ÷ 6
  2. 12x ÷ 4x
  3. 100x ÷ 0
  4. -4x2y ÷ 4x
  5. (3m2 + 3n) ÷ 3
  6. (4n2q + 8m2q) ÷ 4q
  7. (3x × 4x) ÷ 12x2
  8. 14 3 z3 × 6 7 z ÷ 2z
  9. 91x2y2 ÷ (-13x2y2)
  10. -4z2 × (- 3z2) ÷ (-4z)

Worksheet 6

Multiple choice questions

1) The sum of 4a2 and 6a2 is
  1. 10a2
  2. 10a4
  3. 4a4
  4. 6a4
2) 3x2y × 5x2y gives
  1. 15x2y2
  2. 15x4y2
  3. 15x2y
  4. 15xy2
3) 45x3y2 ÷ 15xy gives
  1. 3x2y2
  2. 3x2y
  3. 3xy
  4. 3x3y2
4) 95x2y - 90x2y
  1. 5x4y2
  2. 5x2y2
  3. 5x2y
  4. 5x0y0
5) The result of (7p2 + 8q + 9r) + (3p2 + 2q + r)
  1. 10p2 + 8q + 10r
  2. 10p2 + 6q + 9r
  3. 10p2 + 2q + r
  4. 10(p2 + q + r)
6) (3x2 + 5x - 6) - (10x2 - 5x - 8)
  1. 7x2 + 10x + 14
  2. 7x2 + 5x + 2
  3. -7x2 + 10x + 2
  4. -7x2 + 10x + 14
7) (x + 2)(x + 1)
  1. x2 + 2x + 2
  2. x2 + 3x + 2
  3. x2 + 2x + 1
  4. x2 + 2x + 3
8) (36x2 - 25y2) ÷ (6x - 5y) gives
  1. 6x - 5y
  2. (6x - 5y)2
  3. 6x + 5y
  4. (6x + 5y)2
9) The sides of a triangle are x + 2, x + 5 and 2x + 3, its perimeter is
  1. 4x + 10
  2. 4x + 5
  3. 4x + 7
  4. 4x + 2
10) The each four sides of a square are (x + 1) cm, then its area is
  1. 4(x + 1) cm2
  2. 4(x + 1)2 cm2
  3. (x + 1)2 cm2
  4. (x + 1) cm2
Last updated on: Oct 13, 2025

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