Quick Summary
The Algebra chapter in Class 8 builds on basic expressions to introduce powerful algebraic identities and factorisation techniques. These skills are the foundation for higher algebra in Classes 9 and 10 and carry significant marks in all term exams.
Key Definitions
- Algebraic Expression: A combination of variables, constants, and operations. Example: 3x² + 5x − 7.
- Polynomial: An algebraic expression where powers of variables are non-negative integers.
- Identity: An equation true for all values of the variables.
- Factorisation: Writing an expression as a product of simpler expressions.
Important Algebraic Identities
| Identity | Expanded Form |
|---|---|
| (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 |
Using the Identities — Worked Examples
(a + b)²
Expand (2x + 3)²: = (2x)² + 2(2x)(3) + (3)² = 4x² + 12x + 9
(a − b)²
Expand (5m − 2)²: = (5m)² − 2(5m)(2) + (2)² = 25m² − 20m + 4
(a + b)(a − b)
Expand (y + 7)(y − 7): = y² − 49
Using identities for arithmetic:
- 103² = (100 + 3)² = 10000 + 600 + 9 = 10609
- 98 × 102 = (100 − 2)(100 + 2) = 10000 − 4 = 9996
Factorisation
Method 1 — Common Factor
6x² + 9x = 3x(2x + 3)
Method 2 — Grouping
ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
Method 3 — Using Identities
x² + 10x + 25 = (x + 5)² m² − 16 = (m + 4)(m − 4)
Important Questions
One Mark
- Expand (a + 4)² using an identity.
- Factorise: x² − 9.
- Write the identity for (a − b)².
- Evaluate 99² using the identity (100 − 1)².
- Find the product (x + 5)(x − 5).
Short Answer (3 marks)
- Expand and simplify: (3x + 2)² − (3x − 2)².
- Factorise: 4a² − 12ab + 9b².
- Using identity, find the value of 97 × 103.
Long Answer (5 marks)
- Factorise completely: 2x³ + 4x² − 6x, showing each step.
- Expand (2a + 3b)² and (2a − 3b)² and find their sum and difference. Interpret the result using identities.
Common Mistakes
- Forgetting the middle term (2ab) when expanding (a ± b)².
- Applying (a + b)² = a² + b² (wrong — the 2ab term is omitted).
- Not checking if the answer can be further simplified after factorisation.
Revision Checklist
- [ ] I have memorised all four standard identities.
- [ ] I can expand expressions using identities without a calculator.
- [ ] I can factorise by common factor, grouping, and identities.
- [ ] I have used identities to solve numerical problems (e.g., 103²).
- [ ] I have solved all textbook exercises and corrected mistakes.
Frequently Asked Questions
(a+b)² = a² + 2ab + b², (a−b)² = a² − 2ab + b², and (a+b)(a−b) = a² − b². These must be memorised as they appear in almost every exam.