CBSE · Class 9
Algebra Formulas for Class 9
All algebraic identities, polynomial formulas, and linear equation methods for CBSE Class 9 Maths
Ch 1.Algebraic Identities
| Formula / Relation | Expression |
|---|---|
| (a + b)² | (a + b)² = a² + 2ab + b² |
| (a − b)² | (a − b)² = a² − 2ab + b² |
| (a + b)(a − b) | (a + b)(a − b) = a² − b² |
| (a + b + c)² | (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca |
| (a + b)³ | (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) |
| (a − b)³ | (a − b)³ = a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a − b) |
| a³ + b³ | a³ + b³ = (a + b)(a² − ab + b²) |
| a³ − b³ | a³ − b³ = (a − b)(a² + ab + b²) |
| a³ + b³ + c³ − 3abc | a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca) |
Ch 2.Polynomials
| Formula / Relation | Expression |
|---|---|
| Remainder theorem | If p(x) divided by (x − a), remainder = p(a) |
| Factor theorem | (x − a) is a factor of p(x) ⟺ p(a) = 0 |
| Degree of polynomial | Highest power of variable in the polynomial |
| Zero of polynomial | Value of x where p(x) = 0 |
| Relationship: zeros and coefficients (quadratic) | Sum of zeros = −b/a; Product of zeros = c/a |
Ch 3.Linear Equations in Two Variables
| Formula / Relation | Expression |
|---|---|
| Standard form | ax + by + c = 0 |
| Slope-intercept form | y = mx + c |
| Slope formula | m = (y₂ − y₁) / (x₂ − x₁) |
| Equation of x-axis | y = 0 |
| Equation of y-axis | x = 0 |
Ch 4.Coordinate Geometry
| Formula / Relation | Expression |
|---|---|
| Distance formula | d = √[(x₂−x₁)² + (y₂−y₁)²] |
| Midpoint formula | M = ((x₁+x₂)/2, (y₁+y₂)/2) |
| Section formula (internal division) | P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)) |
| Quadrant signs | Q1:(+,+), Q2:(−,+), Q3:(−,−), Q4:(+,−) |
Ch 5.Number Systems (Surds and Exponents)
| Formula / Relation | Expression |
|---|---|
| Exponent rules | aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ |
| Zero exponent | a⁰ = 1 (a ≠ 0) |
| Negative exponent | a⁻ⁿ = 1/aⁿ |
| Rationalisation | √a × √a = a; 1/√a = √a/a |
| Conjugate rationalisation | 1/(√a + √b) = (√a − √b)/(a − b) |
Frequently Asked Questions
What are the most important algebra identities for Class 9?
The most exam-important identities are: (a+b)², (a−b)², a²−b², (a+b)³, (a−b)³, a³+b³, a³−b³, and the trinomial identity a³+b³+c³−3abc. Master all 8 identities — they appear in direct questions and as tools in complex problems throughout Class 9 and 10.
How do you factorise a polynomial using the factor theorem?
Step 1: Find a value 'a' such that p(a) = 0. This means (x − a) is a factor. Step 2: Divide p(x) by (x − a) using long division or synthetic division. Step 3: Factorise the quotient further if possible. For cubic polynomials, try a = ±1, ±2, ±3 first (factors of the constant term).
What is the distance formula in Class 9?
Distance between two points (x₁,y₁) and (x₂,y₂): d = √[(x₂−x₁)² + (y₂−y₁)²]. This comes from the Pythagorean theorem applied to the horizontal and vertical distances. Special case: distance from origin to point (x,y) = √(x²+y²).
How to remember all algebraic identities for Class 9?
Group them: (1) Square identities: (a+b)², (a−b)², (a+b)(a−b) — these three are related; (2) Cube identities: (a+b)³, (a−b)³ — these follow the binomial pattern; (3) Sum/difference of cubes: a³+b³, a³−b³ — both factorise into two brackets; (4) The 3abc identity. Practice writing them out daily for a week — that's faster than any trick.
What is the remainder theorem in simple terms?
Remainder theorem: When you divide polynomial p(x) by (x − a), the remainder is simply p(a). For example, to find the remainder when x³ − 2x + 1 is divided by (x − 2), just calculate p(2) = 8 − 4 + 1 = 5. The remainder is 5. If p(a) = 0, then (x − a) is a factor (factor theorem).
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