Algebra
ALGEBRA
Introduction to Algebra
| Concept | Explanation | Example |
|---|---|---|
| Variable | Symbol representing unknown value | x, y, a |
| Term | Single part of an expression | 3x, 5y |
| Coefficient | Number multiplying a variable | In 4x, coefficient is 4 |
| Constant | Fixed value | 7, -2 |
| Type of Expression | Example |
|---|---|
| Monomial | 5x |
| Binomial | x + 3 |
| Trinomial | x² + 2x + 1 |
Letters are used to represent unknown or varying values in algebra.
Always simplify expressions by combining like terms.
Expressions and Operations
| Operation | Rule | Example |
|---|---|---|
| Addition | Add like terms only | 3x + 2x = 5x |
| Multiplication | Multiply coefficients and variables | 2x × 3x = 6x² |
| Exponents | Add powers when bases are same | x² × x³ = x⁵ |
| Identity | Formula |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a – b)² | a² – 2ab + b² |
| (a + b)(a – b) | a² – b² |
Negative exponent: a⁻ⁿ = 1 / aⁿ
Use standard identities to solve quickly in exams.
Equations and Their Applications
| Type | Description |
|---|---|
| Linear Equation | Equation of degree 1 |
| Pair of Linear Equations | Two equations with two variables |
| Method | Solution Type |
|---|---|
| Elimination | Remove one variable |
| Substitution | Substitute one equation into another |
| Cross-multiplication | Solve directly |
Nature of roots depends on discriminant D = b² − 4ac.
If D > 0 → Two distinct roots, D = 0 → Equal roots, D < 0 → No real roots.
Factorization and Division
| Method | Example |
|---|---|
| Common Factor | 6x + 12 = 6(x + 2) |
| Identity Method | x² – 9 = (x – 3)(x + 3) |
| Splitting Middle Term | x² + 5x + 6 = (x + 2)(x + 3) |
| Division Type | Example |
|---|---|
| Polynomial ÷ Monomial | (6x² ÷ 3x) = 2x |
| Polynomial ÷ Polynomial | x² + 3x ÷ x |
Dividend = Divisor × Quotient + Remainder
Always arrange polynomials in descending powers before dividing.
Polynomials
| Degree | Polynomial Example |
|---|---|
| 1 | 3x + 2 |
| 2 | x² + 5x + 6 |
| 3 | x³ – 4x² + x |
| Concept | Explanation |
|---|---|
| Zeroes | Value for which polynomial becomes zero |
| Graph | Visual representation of polynomial |
For linear polynomial ax + b, zero = -b/a
Sign of coefficient affects direction of the graph.
Linear Graphs
| Form | Equation |
|---|---|
| Straight line | y = mx + c |
| Horizontal line | y = k |
| Vertical line | x = k |
Slope (m) represents steepness of the line.
If slope is positive, line rises left to right.
Progressions
| Progression | Formula |
|---|---|
| AP nth term | aₙ = a + (n − 1)d |
| AP Sum | Sₙ = n/2 [2a + (n − 1)d] |
| GP nth term | aₙ = arⁿ⁻¹ |
AP → Common difference constant, GP → Common ratio constant.
In AP, difference between consecutive terms is constant.