Algebraic Expressions and Identities (original) (raw)

Last Updated : 23 Jul, 2025

**An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations, representing a value without an equality sign. Whereas, **algebraic identities are equations that hold true for all values of the variables involved. Learning different algebraic identities is crucial for solving equations and simplifying complex problems in mathematics.

To understand these terms, we need to have an understanding of terms, factors, and coefficients. In this article, we have discussed **algebraic expressions and identities with their examples in detail.

Table of Content

**Algebraic Expressions and Identities Definition

An **algebraic expression is a combination of constants, variables, and operations (such as addition, subtraction, multiplication, and division) linked together by mathematical operators. Examples include 3x + 5, 2a2 + 4b - 7, or 5x - 3y + z. These expressions do not have an equality sign.

An **algebraic identity is a mathematical equation that holds true for all possible values of the variables within it. Unlike equations that may have specific solutions, identities are universally valid and help simplify expressions and solve complex problems.

Algebraic Expressions

**Algebraic Expressions are combinations of variables, constants, and arithmetic operations (addition, subtraction, multiplication, division, and exponentiation). They are used to represent mathematical relationships and can be simplified or evaluated for specific values of the variables. An algebraic expression does not have an equality sign, differentiating it from an equation.

Components of Algebraic Expressions

**Terms: In algebra, a term can be a variable a constant, or a constant multiplied by a variable.

**Example: 3x, 4, xy.

Factors: In algebra, factors are all the possible parts of the product.

**Note: 1 is a factor for everything.
**Example 1: Factors of 5x are 1,5,x, and 5x.
**Example 2: Factors of 6x(y+7) are 1,6,x, y+7.

**Coefficients: In algebra, When a term is formed when a constant is multiplied by a variable or variables, that constant is called a coefficient.

**Example1: 5x: In this term, 5 is the coefficient.
**Reason: As 5 is a constant and is being multiplied to a variable 'x' by definition '5' is called a coefficient.

**Example 2: 3x+4y: In this expression, 3, 4 are coefficient.
**Reason: s 3 and 4 are constants and are being multiplied to a variable 'x' and 'y' so by definition '3,4' are called as coefficients.

**Expressions in Algebra

An algebraic expression is an expression that is made up of variables and constants, along with algebraic operations (like subtraction, addition, multiplication, etc.). Expressions are made up of terms.

**Example: 5x+20y, 6-8x.

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**Types of Algebraic Expressions

Expressions in Algebra are divided into three types based on the number of Terms involved in the expression. These types are:

**1. Monomial Expression

Algebraic expressions that contain only one term are called Monomial Expressions.
**Examples: 5x, 10y, 25yz, etc.

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**2. Binomial Expression

Algebraic expression which has two terms (different or unlike terms) is called binomial expression.
**Examples: 30xy+60, 25x+24y, 7+8yz, etc.

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**3. Polynomial Expression

Algebraic expression which contains more than one term with non-negative integer exponents is called Polynomial Expression.
**Examples: 2x+3y+4z, 10x+20y+45,etc.

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**Algebraic Identities

**Algebraic Identities are equations that hold true for all values of the variables involved. They are often used to simplify expressions and solve equations. Identifying and applying these identities can make complex algebraic manipulations more manageable.

**Algebraic Identities List

Some of the most common Algebraic Identities are given in the following list:

**Identity 1: (a + b) 2 = a 2 + 2ab + b 2

**Identity 2: (a – b) 2 = a 2 – 2ab + b 2

**Identity 3: a 2 – b 2 = (a + b)(a – b)

**Identity 4: (x + a)(x + b) = x 2 + (a + b) x + ab

**Identity 5: (a + b + c) 2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ca

**Identity 6: (a + b) 3 = a 3 + b 3 + 3ab (a + b)

**Identity 7: (a – b) 3 = a 3 – b 3 – 3ab (a – b)

**Identity 8: a 3 + b 3 + c 3 **– 3abc = (a + b + c)(a 2 + b 2 + c 2 – ab – bc – ca)

**Identity 9: (x + a) (x + b) = x 2 **+ (a + b)x + ab

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**Example: Implementthe firstIdentity on x = 4, and y = 3

**Solution:

Applying the identity:
L.H.S => (x+y)2 = (4+3)2
= (7)2
= 49

R.H.S => 42 +32+ 2 (4) (3) = 49

As L.H.S = R.H.S, this identity is verified and true.

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**Example: Implement the second Identity on x = 4, and y = 3

**Solution:

Applying the identity:
L.H.S => (x - y)2 = (4 - 3)2
= (1)2
= 1

R.H.S => 42 + 32 - 2 (4) (3) = 1

As L.H.S = R.H.S, this identity is verified and true.

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**Example: Implement the third Identity on x = 4, and y = 3

**Solution:

Applying the identity:
L.H.S => (x + y)(x - y) = (4 + 3)(4 - 3)
= (7)(1)
= 7

R.H.S : x2 - y2 = (4)2 - (3)2

= 16 - 9

= 7

As L.H.S = R.H.S, this identity is verified and true.

Algebraic-Expressions-and-Identities-3

**Example: Implement the fourth Identity on x = 3, y = 4, and z = 5.

**Solution:

Applying the identity:

L.H.S => (x + y + z)2 = (3 + 4 + 5)2

= (12) 2

= 144

R.H.S => x2 - y2 = (3)2 + (4)2 + (5)2 + 2(3)(4) + 2(4)(5) + 2(5)(3)
= 144

As L.H.S = R.H.S, this identity is verified and true.

Using the above Identities we can derive many identities, some of the popularly used identities are written below:

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Algebraic Identities Examples

**1. Simplify (3x + 4) 2

(3x + 4)^2 = (3x)^2 + 2 \cdot 3x \cdot 4 + 4^2 = 9x^2 + 24x + 16

**2. Factorize: x 2 - 16

x^2 - 16 = (x + 4)(x - 4)

**3. Expand (2a - 3b) 3

(2a - 3b)^3 = (2a)^3 - 3 \cdot (2a)^2 \cdot 3b + 3 \cdot 2a \cdot (3b)^2 - (3b)^3 = 8a^3 - 36a^2b + 54ab^2 - 27b^3

**4. **A square garden has a side length of 3x+2 meters. Write an expression for the area of the garden and simplify it using algebraic identities.

**Solution:

The area A of a square is given by the formula side2.
Let the side length be 3x+2

The area is:

A=(3x+2)2

Using the identity (a + b)^2 = a^2 + 2ab + b^2, where a=3x and b=2,

A = (3x)^2 + 2(3x)(2) + 2^2 = 9x^2 + 12x + 4

**Thus, the area of the square garden is 9x^2 + 12x + 4 square meters

**5. The length of a rectangle is 5 meters more than its width. Write an expression for the area of the rectangle in terms of its width, and simplify it using algebraic identities.

**Solution:

Let the width of the rectangle be x meters.
The length of the rectangle is x+5 meters.

The area A of a rectangle is given by length × width.

A=(x+5)×x

Now, expand using the distributive property:

A=x2+5x

Thus, the area of the rectangle is x2+5x square meters.

Algebraic Identities Worksheet

**Question 1: Simplify the expression using algebraic identities: (4x + 7)^2

**Question 2: Factorize the expression using algebraic identities: x^2 - 16

**Question 3: Simplify the expression using algebraic identities: (3x - 5)(3x + 5)

**Question 4: Simplify the expression: (x + 4)(x + 6)

**Question 5: Factorize the expression using algebraic identities: 9x^2 - 25

**Question 6: Expand the expression using algebraic identities: (x - 3)^2

**Question 7: Simplify the expression using algebraic identities: (2x + 5)(2x - 5)

**Question 8: Factorize the expression: 49y^2 - 36

**Question 9: Expand the expression: (5x - 2)^2

**Question 10: Simplify the expression using algebraic identities: (x - 6)(x + 8)

Answer Key

  1. (4x + 7)^2 = 16x^2 + 56x + 49
  2. x^2 - 16 = (x - 4)(x + 4)
  3. (3x - 5)(3x + 5) = 9x^2 - 25
  4. (x + 4)(x + 6) = x^2 + 10x + 24
  5. 9x^2 - 25 = (3x - 5)(3x + 5)
  6. (x - 3)^2 = x^2 - 6x + 9
  7. (2x + 5)(2x - 5) = 4x^2 - 25
  8. 49y^2 - 36 = (7y - 6)(7y + 6)
  9. (5x - 2)^2 = 25x^2 - 20x + 4
  10. (x - 6)(x + 8) = x^2 + 2x - 48

Conclusion

**Algebraic expressions and **algebraic identities are essential tools in mathematics, providing a foundation for simplifying complex problems and solving equations. Understanding these concepts allows for greater flexibility in manipulating mathematical statements and helps build problem-solving skills applicable in various fields, from engineering to economics.