Algebra, Factorisation, Equations & Proportion
1. Algebraic Expressions and Identities
Multiplying Polynomials
- Monomial × Polynomial: Use distributive law. 3x(2x² + 4) = 6x³ + 12x.
- Binomial × Binomial: (x + a)(x + b) = x² + (a+b)x + ab
- Binomial × Trinomial: Use distributive law for each term.
Standard Identities
| Identity |
|---|
| (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 |
2. Factorisation
What Is Factorisation?
Writing an algebraic expression as a PRODUCT of its FACTORS. 'The REVERSE of expansion.'
Methods
| Method | When to Use |
|---|---|
| Taking common factor | ALL terms share a factor. 6x + 9 = 3(2x + 3). |
| Grouping | Group terms. Factor each group. Look for common binomial. |
| Using identities | Expression matches a standard identity. x² — 16 = x² — 4² = (x+4)(x-4). |
| Splitting middle term (quadratics) | For x² + bx + c: find two numbers that MULTIPLY to c and ADD to b. x² + 5x + 6 = (x+2)(x+3). |
3. Linear Equations in One Variable
Solving — Step by Step
- Clear denominators (multiply by LCM)
- Remove brackets
- Transpose: variables to one side, constants to other
- Solve for the variable
- CHECK by substituting back
Equations Reducible to Linear Form
Cross-multiplication: (x+1)/(x-1) = 3/2 → 2(x+1) = 3(x-1) → 2x+2 = 3x-3 → x = 5.
Word Problems
- Age problems. Digit problems. Fraction problems. 'Let the unknown be x. Frame the equation. Solve. Check for reasonableness.'
4. Percentage and Its Applications
Profit, Loss and Discount
- Profit = SP — CP. Profit% = (Profit/CP)×100.
- Discount = Marked Price — Selling Price. Discount% = (Discount/MP)×100.
- Successive Discounts: NOT additive. Two 10% discounts ≠ 20%. Final price = Original × 0.9 × 0.9 = 81%.
Tax (GST — Goods and Services Tax)
GST is added to the selling price. SP + GST = Final price paid.
5. Direct and Inverse Proportion
| Direct Proportion | Inverse Proportion |
|---|---|
| x ↑ → y ↑ (or x ↓ → y ↓) | x ↑ → y ↓ (or x ↓ → y ↑) |
| y/x = k (CONSTANT ratio). y = kx. | xy = k (CONSTANT product). y = k/x. |
| More people → more food needed. More time → more distance covered. | More workers → less time. Faster speed → less time. |
6. Compound Interest
Simple vs Compound
- Simple Interest: SI = (P×R×T)/100. Same interest each year.
- Compound Interest: Interest COMPOUNDS — each year's interest is ADDED to the principal for NEXT year's calculation.
Formula
- CI = P(1 + r/100)ⁿ — P (for annual compounding)
- For half-yearly: rate = r/2. Number of periods = 2n.
- For quarterly: rate = r/4. Number of periods = 4n.
Growth and Depreciation
- Population growth: Popₙ = P₀(1 + r/100)ⁿ (same formula as CI)
- Depreciation of value: Valueₙ = P₀(1 — r/100)ⁿ
