- An algebraic expression combines variables and numbers. An identity is an equality true for every permissible value.
- An equation may hold only for particular values; an identity holds for all allowed values.
- a(b+c)=ab+ac. It is the basis for expanding brackets.
- (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.
- (a+b+c)²=a²+b²+c²+2ab+2bc+2ca.
- (a+b)³=a³+3a²b+3ab²+b³.
- (a−b)³=a³−3a²b+3ab²−b³.
- a³+b³=(a+b)(a²−ab+b²).
- a³−b³=(a−b)(a²+ab+b²).
Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.
Exploring Algebraic Identities — Class 9 Mathematics
Edition: Ganita Manjari 2026–27, Part I, Chapter 4. This chapter belongs to the 2026–27 Ganita Manjari sequence.
Direct explanation: An algebraic expression combines variables and numbers. An identity is an equality true for every permissible value.
This is an original expanded study guide. Exact NCERT exercise numbers, diagrams and activities have not been independently verified against a complete official textbook PDF. The practice below is original, not a claimed reproduction of official exercises.
Learning objectives
- Explain definitions and conditions precisely.
- Use mathematical notation and diagrams correctly.
- Show full reasoning in solved problems.
- Check answers using substitution, estimation or geometry.
Expression and identity
An algebraic expression combines variables and numbers. An identity is an equality true for every permissible value.
Worked example
(x+1)²=x²+2x+1 is an identity.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Equation versus identity
An equation may hold only for particular values; an identity holds for all allowed values.
Worked example
x+3=7 has solution x=4, not every x.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Distributive law
a(b+c)=ab+ac. It is the basis for expanding brackets.
Worked example
4(x+3)=4x+12.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Square of a sum
(a+b)²=a²+2ab+b².
Worked example
(x+5)²=x²+10x+25.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Square of a difference
(a−b)²=a²−2ab+b².
Worked example
(2x−3)²=4x²−12x+9.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Product of sum and difference
(a+b)(a−b)=a²−b².
Worked example
(x+7)(x−7)=x²−49.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Two binomials
(x+a)(x+b)=x²+(a+b)x+ab.
Worked example
(x+2)(x+5)=x²+7x+10.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Square of three terms
(a+b+c)²=a²+b²+c²+2ab+2bc+2ca.
Worked example
(x+y+2)²=x²+y²+4+2xy+4x+4y.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Cube of a sum
(a+b)³=a³+3a²b+3ab²+b³.
Worked example
(x+2)³=x³+6x²+12x+8.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Cube of a difference
(a−b)³=a³−3a²b+3ab²−b³.
Worked example
(x−1)³=x³−3x²+3x−1.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Sum of cubes
a³+b³=(a+b)(a²−ab+b²).
Worked example
x³+8=(x+2)(x²−2x+4).
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Difference of cubes
a³−b³=(a−b)(a²+ab+b²).
Worked example
x³−27=(x−3)(x²+3x+9).
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Identity a³+b³+c³−3abc
a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca).
Worked example
If a+b+c=0, then a³+b³+c³=3abc.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Factorisation
Reverse expansion to express a polynomial as a product of simpler factors.
Worked example
x²+6x+9=(x+3)².
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Geometric verification
Area diagrams can illustrate square identities by partitioning a large square into smaller regions.
Worked example
A square of side a+b has area a²+2ab+b².
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Error checks
The middle term in (a+b)² is 2ab, not ab; signs in cubic identities must be tracked carefully.
Worked example
(x−2)²=x²−4x+4, not x²−4.
Concept check
Explain why the example satisfies the stated definition or formula and identify any conditions that must hold.
Formula and concept reference
| Concept | Rule | Worked illustration |
|---|---|---|
| Expression and identity | An algebraic expression combines variables and numbers. An identity is an equality true for every permissible value. | (x+1)²=x²+2x+1 is an identity. |
| Equation versus identity | An equation may hold only for particular values; an identity holds for all allowed values. | x+3=7 has solution x=4, not every x. |
| Distributive law | a(b+c)=ab+ac. It is the basis for expanding brackets. | 4(x+3)=4x+12. |
| Square of a sum | (a+b)²=a²+2ab+b². | (x+5)²=x²+10x+25. |
| Square of a difference | (a−b)²=a²−2ab+b². | (2x−3)²=4x²−12x+9. |
| Product of sum and difference | (a+b)(a−b)=a²−b². | (x+7)(x−7)=x²−49. |
| Two binomials | (x+a)(x+b)=x²+(a+b)x+ab. | (x+2)(x+5)=x²+7x+10. |
| Square of three terms | (a+b+c)²=a²+b²+c²+2ab+2bc+2ca. | (x+y+2)²=x²+y²+4+2xy+4x+4y. |
| Cube of a sum | (a+b)³=a³+3a²b+3ab²+b³. | (x+2)³=x³+6x²+12x+8. |
| Cube of a difference | (a−b)³=a³−3a²b+3ab²−b³. | (x−1)³=x³−3x²+3x−1. |
| Sum of cubes | a³+b³=(a+b)(a²−ab+b²). | x³+8=(x+2)(x²−2x+4). |
| Difference of cubes | a³−b³=(a−b)(a²+ab+b²). | x³−27=(x−3)(x²+3x+9). |
| Identity a³+b³+c³−3abc | a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca). | If a+b+c=0, then a³+b³+c³=3abc. |
| Factorisation | Reverse expansion to express a polynomial as a product of simpler factors. | x²+6x+9=(x+3)². |
| Geometric verification | Area diagrams can illustrate square identities by partitioning a large square into smaller regions. | A square of side a+b has area a²+2ab+b². |
| Error checks | The middle term in (a+b)² is 2ab, not ab; signs in cubic identities must be tracked carefully. | (x−2)²=x²−4x+4, not x²−4. |
Exercise A — Definitions and foundational skills
- Define expression and identity and give a mathematically valid example of your own.
- Define equation versus identity and give a mathematically valid example of your own.
- Define distributive law and give a mathematically valid example of your own.
- Define square of a sum and give a mathematically valid example of your own.
- Define square of a difference and give a mathematically valid example of your own.
- Define product of sum and difference and give a mathematically valid example of your own.
- Define two binomials and give a mathematically valid example of your own.
- Define square of three terms and give a mathematically valid example of your own.
- Define cube of a sum and give a mathematically valid example of your own.
- Define cube of a difference and give a mathematically valid example of your own.
Exercise B — Worked-example practice
- Rework this example: (x+1)²=x²+2x+1 is an identity. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: x+3=7 has solution x=4, not every x. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: 4(x+3)=4x+12. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: (x+5)²=x²+10x+25. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: (2x−3)²=4x²−12x+9. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: (x+7)(x−7)=x²−49. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: (x+2)(x+5)=x²+7x+10. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: (x+y+2)²=x²+y²+4+2xy+4x+4y. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: (x+2)³=x³+6x²+12x+8. Then change one numerical value and solve the new problem, showing every step.
- Rework this example: (x−1)³=x³−3x²+3x−1. Then change one numerical value and solve the new problem, showing every step.
Exercise C — Reasoning and challenge
- Prove or justify one central identity or rule in the chapter.
- Give a counterexample to a common false statement.
- Compare two concepts and explain how they differ.
- Create a realistic problem and explain your assumptions.
- Find and correct an intentional arithmetic or reasoning error.
- Describe how a diagram, table or graph supports a conclusion.
- Explain a result to a beginner in five clear sentences.
- Check a result using a different valid method.
Frequently asked questions with answers
How do you understand expression and identity?
An algebraic expression combines variables and numbers. An identity is an equality true for every permissible value. Example: (x+1)²=x²+2x+1 is an identity.
How do you understand equation versus identity?
An equation may hold only for particular values; an identity holds for all allowed values. Example: x+3=7 has solution x=4, not every x.
How do you understand distributive law?
a(b+c)=ab+ac. It is the basis for expanding brackets. Example: 4(x+3)=4x+12.
How do you understand square of a sum?
(a+b)²=a²+2ab+b². Example: (x+5)²=x²+10x+25.
How do you understand square of a difference?
(a−b)²=a²−2ab+b². Example: (2x−3)²=4x²−12x+9.
How do you understand product of sum and difference?
(a+b)(a−b)=a²−b². Example: (x+7)(x−7)=x²−49.
How do you understand two binomials?
(x+a)(x+b)=x²+(a+b)x+ab. Example: (x+2)(x+5)=x²+7x+10.
How do you understand square of three terms?
(a+b+c)²=a²+b²+c²+2ab+2bc+2ca. Example: (x+y+2)²=x²+y²+4+2xy+4x+4y.
How do you understand cube of a sum?
(a+b)³=a³+3a²b+3ab²+b³. Example: (x+2)³=x³+6x²+12x+8.
How do you understand cube of a difference?
(a−b)³=a³−3a²b+3ab²−b³. Example: (x−1)³=x³−3x²+3x−1.
How do you understand sum of cubes?
a³+b³=(a+b)(a²−ab+b²). Example: x³+8=(x+2)(x²−2x+4).
How do you understand difference of cubes?
a³−b³=(a−b)(a²+ab+b²). Example: x³−27=(x−3)(x²+3x+9).
How do you understand identity a³+b³+c³−3abc?
a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca). Example: If a+b+c=0, then a³+b³+c³=3abc.
How do you understand factorisation?
Reverse expansion to express a polynomial as a product of simpler factors. Example: x²+6x+9=(x+3)².
How do you understand geometric verification?
Area diagrams can illustrate square identities by partitioning a large square into smaller regions. Example: A square of side a+b has area a²+2ab+b².
Revision checklist
- I can explain and apply expression and identity.
- I can explain and apply equation versus identity.
- I can explain and apply distributive law.
- I can explain and apply square of a sum.
- I can explain and apply square of a difference.
- I can explain and apply product of sum and difference.
- I can explain and apply two binomials.
- I can explain and apply square of three terms.
- I can explain and apply cube of a sum.
- I can explain and apply cube of a difference.
- I can explain and apply sum of cubes.
- I can explain and apply difference of cubes.
- I can explain and apply identity a³+b³+c³−3abc.
- I can explain and apply factorisation.
- I can explain and apply geometric verification.
- I can explain and apply error checks.
Official textbook reference
NCERT official textbook portal. Supplementary notes and original practice exercises; not verified official NCERT question-by-question solutions or a Class 9 board marking scheme.
