CBSE Class 9 Maths Revision Notes Chapter 4 Exploring Algebraic Identities

Algebraic identities are equations that remain true for every permitted value of their variables. They help expand expressions, factor polynomials, simplify calculations and solve area or volume problems efficiently.

The chapter Exploring Algebraic Identities develops important algebraic rules used in expansion and factorisation. It also explains how geometric models and algebra tiles represent expressions visually.

These CBSE Class 9 Maths Revision Notes Chapter 4 cover square identities, cubic identities, splitting the middle term and rational expressions. The notes follow the current 2026–27 textbook.

Key Takeaways

• Identity: An equation true for all values of its variables.
• Perfect-square form: a² ± 2ab + b² = (a ± b)².
• Difference of squares: a² − b² = (a + b)(a − b).
• Factorisation: It rewrites an expression as a product of simpler factors.

Access Class 9 Maths Chapter 4 Notes in 30 Minutes

Use this order for quick revision:

First 10 minutes: Square identities and difference of squares
Next 10 minutes: Factorisation and splitting the middle term
Final 10 minutes: Cubic identities and rational expressions

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Algebraic Identities in Class 9 Maths Chapter 4 Notes

An algebraic identity connects two expressions that have the same value for every allowed value of the variables.

What Is an Algebraic Identity?

An algebraic identity is an equation that remains true for all values of the variables.

For example:

(x + y)² = x² + 2xy + y²

This is true for every value of x and y.

Difference Between an Equation and an Identity

An equation may be true only for particular values. An identity is true for all values.

The equation x² − 1 = 24 is true only when x = 5 or x = −5.

How Can Identities Be Visualised?

Areas of squares and rectangles can represent algebraic terms.

For a square of side a + b:

• Total area = (a + b)²
• Large square area = a²
• Small square area = b²
• Two rectangle areas = ab + ab

Therefore:

(a + b)² = a² + 2ab + b²

Expanding Expressions in CBSE Class 9 Maths Chapter 4 Notes

Expansion means removing brackets and writing an expression as a sum of terms.

Square of a Sum

The identity is:

(a + b)² = a² + 2ab + b²

Remember the order:

1. Square the first term.
2. Add twice the product.
3. Add the square of the second term.

Example 1

Expand:

(5x + 2y)²

Using the identity:

= (5x)² + 2(5x)(2y) + (2y)²

= 25x² + 20xy + 4y²

Example 2

Find 43².

43² = (40 + 3)²

= 40² + 2(40)(3) + 3²

= 1600 + 240 + 9

= 1849

Square of a Difference

The identity is:

(a − b)² = a² − 2ab + b²

Only the middle term is negative.

Example 3

Expand:

(3x − 4y)²

= (3x)² − 2(3x)(4y) + (4y)²

= 9x² − 24xy + 16y²

Example 4

Find 29².

29² = (30 − 1)²

= 30² − 2(30)(1) + 1²

= 900 − 60 + 1

= 841

Square of Three Terms

The identity is:

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Each term is squared. Twice every pairwise product is also included.

Example 5

Expand:

(x + 2y + 3z)²

= x² + 4y² + 9z² + 4xy + 12yz + 6xz

Signs in a Three-Term Square

For an expression such as:

(a − b + c)²

Treat the second term as −b.

= a² + b² + c² − 2ab − 2bc + 2ac

Always use the actual signs of the terms.

Difference of Squares in Class 9 Maths Chapter 4 Notes

The identity is:

a² − b² = (a + b)(a − b)

It applies only when two perfect squares are being subtracted.

Example 6

Factor:

49x² − 25y²

= (7x)² − (5y)²

= (7x + 5y)(7x − 5y)

Numerical Use

Find:

55²

Using:

a² = (a + b)(a − b) + b²

Choose b = 5.

55² = (55 + 5)(55 − 5) + 5²

= 60 × 50 + 25

= 3025

Factorisation in Class 9 Maths Revision Notes Chapter 4

Factorisation means expressing an algebraic expression as a product.

It reverses expansion.

Factorising a Perfect-Square Trinomial

Compare the expression with:

a² + 2ab + b² = (a + b)²

or

a² − 2ab + b² = (a − b)²

Example 7

Factor:

x² + 4x + 4

Here:

• x² = x²
• 4 = 2²
• 4x = 2(x)(2)

Therefore:

x² + 4x + 4 = (x + 2)²

Example 8

Factor:

36x² − 12x + 1

= (6x)² − 2(6x)(1) + 1²

= (6x − 1)²

Taking a Common Factor First

Sometimes the complete expression is not immediately a perfect square.

Example 9

Factor:

50p² + 60pq + 18q²

Take 2 common:

= 2(25p² + 30pq + 9q²)

The expression inside is:

(5p)² + 2(5p)(3q) + (3q)²

Therefore:

50p² + 60pq + 18q² = 2(5p + 3q)²

Product of Two Binomials

The general identity is:

(x + a)(x + b) = x² + (a + b)x + ab

Example 10

Expand:

(x + 3)(x + 4)

= x² + (3 + 4)x + 3 × 4

= x² + 7x + 12

General Product with Coefficients

(px + a)(qx + b) = pqx² + (pb + aq)x + ab

Example 11

Expand:

(2x + 3)(3x + 1)

= 6x² + 2x + 9x + 3

= 6x² + 11x + 3

Splitting the Middle Term Class 9 Notes

A quadratic expression can often be factorised by splitting its middle term.

For:

x² + bx + c

find two numbers whose:

• Sum is b
• Product is c

Example 12

Factor:

x² + 7x + 12

We need two numbers whose sum is 7 and product is 12.

The numbers are 3 and 4.

x² + 7x + 12

= x² + 3x + 4x + 12

= x(x + 3) + 4(x + 3)

= (x + 3)(x + 4)

Example 13

Factor:

x² − 5x + 6

We need two numbers whose sum is −5 and product is 6.

The numbers are −2 and −3.

x² − 5x + 6

= x² − 2x − 3x + 6

= x(x − 2) − 3(x − 2)

= (x − 2)(x − 3)

When the First Coefficient Is Not 1

For:

ax² + bx + c

find two numbers whose:

• Sum is b
• Product is ac

Then split the middle term.

Example 14

Factor:

6x² + 7x + 2

Here:

a × c = 6 × 2 = 12

We need two numbers with sum 7 and product 12.

The numbers are 3 and 4.

6x² + 7x + 2

= 6x² + 3x + 4x + 2

= 3x(2x + 1) + 2(2x + 1)

= (3x + 2)(2x + 1)

Algebra Tiles in Exploring Algebraic Identities Notes

Algebra tiles provide a visual method for representing expressions.

Common tiles include:

• One x² tile
• Rectangular x tiles
• Unit square tiles

For example:

x² + 7x + 12

can be arranged into a rectangle with sides:

x + 3 and x + 4

Therefore:

x² + 7x + 12 = (x + 3)(x + 4)

Algebra tiles show why the middle term is split into 3x + 4x.

Cubic Identities in Class 9 Chapter 4 Maths Notes

Cubic identities contain terms raised to the third power.

Cube of a Sum

(a + b)³ = a³ + 3a²b + 3ab² + b³

Another useful form is:

(a + b)³ = a³ + b³ + 3ab(a + b)

Example 15

Expand:

(x + 2y)³

= x³ + 3x²(2y) + 3x(2y)² + (2y)³

= x³ + 6x²y + 12xy² + 8y³

Cube of a Difference

(a − b)³ = a³ − 3a²b + 3ab² − b³

The signs alternate.

Example 16

Expand:

(2x − 3y)³

= 8x³ − 36x²y + 54xy² − 27y³

Sum and Difference of Cubes

Difference of Cubes

a³ − b³ = (a − b)(a² + ab + b²)

Example 17

Factor:

8x³ − 27y³

= (2x)³ − (3y)³

= (2x − 3y)(4x² + 6xy + 9y²)

Sum of Cubes

a³ + b³ = (a + b)(a² − ab + b²)

Example 18

Factor:

64x³ + 125y³

= (4x)³ + (5y)³

= (4x + 5y)(16x² − 20xy + 25y²)

Sign Pattern

The sign in the first factor matches the original sign. The middle sign in the second factor is opposite.

Three-Variable Cubic Identity

The identity is:

x³ + y³ + z³ − 3xyz

= (x + y + z)(x² + y² + z² − xy − yz − zx)

A useful result follows when:

x + y + z = 0

Then:

x³ + y³ + z³ = 3xyz

Example 19

If a + b + c = 0, then:

a³ + b³ + c³ = 3abc

This is often used in proof-based questions.

Rational Algebraic Expressions in Class 9 Maths Chapter 4 Notes

A rational algebraic expression is a fraction whose numerator and denominator are algebraic expressions.

Example:

(x² − 7x + 12)/(5x² + 5x − 100)

Steps for Simplification

1. Factor the numerator.
2. Factor the denominator.
3. Identify common factors.
4. Cancel only common factors.
5. State that the denominator must not be zero.

Example 20

Simplify:

(x² − 7x + 12)/(5x² + 5x − 100)

Factor the numerator:

x² − 7x + 12 = (x − 3)(x − 4)

Factor the denominator:

5x² + 5x − 100

= 5(x² + x − 20)

= 5(x + 5)(x − 4)

Therefore:

[(x − 3)(x − 4)]/[5(x + 5)(x − 4)]

Cancel the common factor x − 4:

= (x − 3)/[5(x + 5)]

The original denominator must remain non-zero.

Important Cancellation Rule

Terms cannot be cancelled across addition or subtraction.

Incorrect:

(x + 3)/x = 3

Correct cancellation is possible only after factorisation.

Applications of Algebraic Identities

Identities are useful for:

• Calculating squares and cubes.
• Factorising expressions.
• Finding dimensions from areas.
• Simplifying rational expressions.
• Solving area and volume problems.
• Proving numerical patterns.

Area Application

Suppose the area of a rectangle is:

x² + 8x + 15

Factor:

x² + 8x + 15 = (x + 3)(x + 5)

Possible dimensions are:

• Length = x + 5
• Breadth = x + 3

Pool Problem

A rectangular pool has length x metres and breadth x − 4 metres.

Area = 96 m²

x(x − 4) = 96

x² − 4x − 96 = 0

= (x − 12)(x + 8)

So:

x = 12 or x = −8

Length cannot be negative.

Therefore:

• Length = 12 m
• Breadth = 8 m

Algebraic Identity Formula Table

Quick Revision of Exploring Algebraic Identities

• An identity is true for all values of its variables.
• Expansion removes brackets.
• Factorisation writes an expression as a product.
• (a + b)² contains +2ab.
• (a − b)² contains −2ab.
• Difference of squares has two conjugate factors.
• Splitting the middle term helps factor quadratics.
• Algebra tiles give a visual model of factorisation.
• Cubic identities expand or factor cube expressions.
• Sum and difference of cubes use different sign patterns.
• Rational expressions must be factorised before cancellation.
• Denominators must remain non-zero.

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