ZestExam

Algebraic Identities

CRPF Constable · Quantitative Aptitude

Also prepare for:BSF Const CISF Const ITBP Const UP Const

← Quadratic Equations Polynomials →

💡

Core Concept

Read this first — the foundation of the topic

💡Key Rules and Properties

Algebraic identities are universally true for all values of variables. They help expand, factorize, and simplify expressions. The most important ones follow specific patterns that SSC loves to test

🔑Essential Formulas

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

📊

Exam Patterns

What examiners ask — read before attempting PYQs

SSC CGL typically asks direct substitution problems, value finding questions, and simplification using identities. Common question types include finding the value of expressions like x² + 1/x² when x + 1/x is given, or simplifying complex algebraic fractions. Shortcut Trick - The 'Recognition Method': Instead of expanding everything, learn to recognize patterns. If you see (something)² - (something else)², immediately think a² - b² = (a+b)(a-b). This saves 2-3 minutes per question.

✏️

Worked Example

Solve this step-by-step before moving on

1

Step 1

Start with the given condition x + 1/x = 5

2

Step 2

Square both sides: (x + 1/x)² = 5² = 25

3

Step 3

Apply the identity (a + b)² = a² + 2ab + b² Here, a = x and b = 1/x

4

Step 4

(x + 1/x)² = x² + 2(x)(1/x) + (1/x)²

5

Step 5

25 = x² + 2(1) + 1/x²

6

Step 6

25 = x² + 2 + 1/x²

7

Step 7

x² + 1/x² = 25 - 2 = 23 Therefore, x² + 1/x² = 23. Common Mistake: Students often forget the middle term in expansions. In (a + b)², many write a² + b² and miss the 2ab term. Always count terms: square of binomial has THREE terms, cube has FOUR terms. Also, students confuse signs in (a - b) expansions. Remember: alternate signs in (a - b)³.

🔑 Key Points

(a + b)² = a² + 2ab + b² - never forget the middle term 2ab
(a - b)² = a² - 2ab + b² - note the minus sign before 2ab
(a + b)(a - b) = a² - b² - difference of squares identity
a³ + b³ = (a + b)(a² - ab + b²) - sum of cubes factorization
a³ - b³ = (a - b)(a² + ab + b²) - difference of cubes factorization
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca - three variable expansion
When x + 1/x is given, find x² + 1/x² by squaring both sides
Recognize patterns first, then apply appropriate identity to save time

📌 Exam Facts

(a + b)² has exactly 3 terms: a², 2ab, and b²
(a - b)³ = a³ - 3a²b + 3ab² - b³ with alternating signs
x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
If a + b + c = 0, then a³ + b³ + c³ = 3abc
(a + b)³ - (a - b)³ = 2b(3a² + b²)
The coefficient of middle term in (a + b)² is always 2
a⁴ - b⁴ = (a² + b²)(a + b)(a - b)
(x + 1/x)² = x² + 1/x² + 2

🚀 60-Second Revision

Remember: Square of binomial has 3 terms, cube has 4 terms
Formula: (a + b)(a - b) = a² - b² for quick factorization
Trap: Don't forget middle terms in expansions - very common error
Trick: If x + 1/x given, square it to find x² + 1/x²
Pattern: Recognize a² - b² form immediately for difference of squares
Sign Rule: (a - b)³ has alternating positive-negative signs
Quick Check: Count terms in your final answer to avoid missing any

🎯

Test yourself under real exam conditions

A timed CRPF Constable mock shows exactly how Algebraic Identities questions appear in the actual paper — and where you lose marks.

Take Free Mock Test →Practice Panel

✓ Free mock✓ Instant percentile✓ AI Battle Plan after test