Algebraic Expressions & Identities — Class 8

Comprehensive notes covering expressions, polynomials, standard identities, proofs, factorization and practice — mobile-friendly & SEO-optimized.

1. Introduction

An algebraic expression is a combination of numbers, variables and arithmetic operations. Example: 3x² − 5x + 2. Expressions do not have an equality sign. When an expression is set equal to another, it's an equation.

2. Terms, Coefficients & Types

  • Term: a product of numbers and variables (e.g., 7xy).
  • Coefficient: numerical factor of a term (e.g., 7 in 7xy).
  • Like terms: terms with same variable part (e.g., 3x² and −5x²).
  • Monomial: single term (e.g., 4x).
  • Binomial: two terms (e.g., x+2).
  • Polynomial: sum of terms (e.g., 2x³ − x + 5).

3. Operations on Expressions

  1. Addition/Subtraction: combine like terms. Example: (3x²+2x) + (−x²+5) = 2x²+2x+5.
  2. Multiplication: multiply coefficients and add exponents for same bases. Example: x² × x³ = x⁵.
  3. Division: divide coefficients and subtract exponents. Example: x⁵ / x² = x³.
  4. Substitution: evaluate expression by substituting values for variables.

4. Standard Algebraic Identities

1. (a + b)² = a² + 2ab + b²
2. (a − b)² = a² − 2ab + b²
3. a² − b² = (a − b)(a + b)
4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
5. (a + b)³ = a³ + 3a²b + 3ab² + b³
6. (a − b)³ = a³ − 3a²b + 3ab² − b³

It's helpful to memorize these identities — they speed up expansion, simplification and factorization.

5. Factorization using Identities

Use identities to factor expressions quickly. Examples:

  • a² − b² = (a − b)(a + b). Example: x² − 9 = (x − 3)(x + 3).
  • a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).
  • (a + b)² − (a − b)² = 4ab (useful trick).

6. Solved Examples

Example 1: Expand (x + 5)².
Using identity: = x² + 10x + 25.
Example 2: Factorize x² − 16.
= (x − 4)(x + 4) (difference of squares).
Example 3: Simplify (a + b)³ − (a − b)³.
Using cube expansions: result = 6ab(a + b). (Show steps in answer.)
Example 4: If x + y = 6 and xy = 8, find x² + y².
x² + y² = (x + y)² − 2xy = 36 − 16 = 20.

7. Practice Questions (with answers)

  1. Expand (2x − 3)².
    Answer
    = 4x² − 12x + 9
  2. Factorize 27y³ − 8.
    Answer
    = (3y − 2)(9y² + 6y + 4)
  3. Simplify (x + y)² − (x − y)².
    Answer
    = 4xy
  4. Given a − b = 3 and a² − b² = 21, find a + b.
    Answer
    Use a² − b² = (a − b)(a + b) → 21 = 3(a + b) → a + b = 7

Write full steps in exams. Use identities for faster expansion and cleaner factorization.