AlGebra NCERT Formulas

Class 8 Algebra Formulas

Basic Algebraic Identities

Formula	Expansion
(a + b)²	a² + 2ab + b²
(a – b)²	a² – 2ab + b²
(a + b)(a – b)	a² – b²
(a + b + c)²	a² + b² + c² + 2ab + 2bc + 2ca
(a – b – c)²	a² + b² + c² – 2ab + 2bc – 2ca

Cube Formulas

Formula	Factorization
(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)

Higher Power Formulas

• (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
• (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴
• a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
• a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)
• aⁿ – bⁿ = (a – b)(aⁿ⁻¹ + aⁿ⁻²b + … + bⁿ⁻¹)

Laws of Exponents

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• (ab)ᵐ = aᵐ × bᵐ
• a⁰ = 1
• a⁻ᵐ = 1/aᵐ

Algebraic Properties

Commutative Property

• Addition: a + b = b + a
• Multiplication: a × b = b × a

Associative Property

• Addition: (a + b) + c = a + (b + c)
• Multiplication: (a × b) × c = a × (b × c)

Distributive Property

• a × (b + c) = a × b + a × c
• a × (b – c) = a × b – a × c

Identity Element

• Addition: a + 0 = a
• Multiplication: a × 1 = a

Inverse Element

• Addition: a + (-a) = 0
• Multiplication: a × (1/a) = 1 (where a ≠ 0)

Class 9 Algebra Formulas

Logarithmic Formulas

• logₐ(xy) = logₐx + logₐy
• logₐ(x/y) = logₐx – logₐy
• logₐ(xᵐ) = m logₐx
• logₐa = 1
• logₐ1 = 0

Class 10 Algebra Formulas

Quadratic Formula

For ax² + bx + c = 0:

• x = [−b ± √(b² − 4ac)] / 2a

Arithmetic Sequence Formulas

For sequence: a, a + d, a + 2d, …

• nth term: aₙ = a + (n – 1)d
• Sum of first n terms: Sₙ = n/2 [2a + (n – 1)d]

Geometric Sequence Formulas

For sequence: a, ar, ar², …

• nth term: aₙ = arⁿ⁻¹
• Sum of first n terms: Sₙ = a(1 – rⁿ)/(1 – r)
• Sum of infinite terms (when |r| < 1): S = a/(1 – r)

Class 11 Algebra Formulas

Factorial Formula

• n! = n × (n – 1) × (n – 2) × … × 3 × 2 × 1

Permutation Formulas

• ⁿPᵣ = n!/(n – r)!

Combination Formula

• ⁿCᵣ = n!/[r!(n – r)!]

Difference of Powers

When n is even:

• aⁿ – bⁿ = (a – b)(aⁿ⁻¹ + aⁿ⁻²b + … + bⁿ⁻¹)

When n is odd:

• aⁿ + bⁿ = (a + b)(aⁿ⁻¹ – aⁿ⁻²b + … + bⁿ⁻¹)

Exponent Rules

• (aᵐ)(aⁿ) = aᵐ⁺ⁿ
• (ab)ᵐ = aᵐbᵐ
• (aᵐ)ⁿ = aᵐⁿ

Class 12 Algebra Formulas

Vector Algebra Formulas

For vector a = xi + yj + zk:

• Magnitude: |a| = √(x² + y² + z²)
• Unit vector: â = a/|a|

Dot Product

• a ⋅ b = |a||b|cosθ (where θ is the angle between vectors)

Cross Product

• a × b = |a||b|sinθ (where θ is the angle between vectors)

Scalar Triple Product

• [a b c] = a ⋅ (b × c) = (a × b) ⋅ c

Vector Addition Properties

• a + b = b + a (Commutative)
• a + (b + c) = (a + b) + c (Associative)
• k(a + b) = ka + kb (Distributive)

Direction Cosines

• For direction angles α, β, γ: l = cosα, m = cosβ, n = cosγ
• cos²α + cos²β + cos²γ = 1

Important Unit Vectors

• î =​
• ĵ =​
• k̂ =​