Algebra : SGT
Algebra
Patterns and Introduction to Algebra
| Concept | Key Idea | Example |
|---|---|---|
| Patterns | Recognizing number or shape patterns | 2, 4, 6, 8 → Rule: +2 |
| Variables | Letters representing numbers | x + 5 = 12 |
| Algebraic Expression | Combination of numbers and variables | 3x + 4 |
| Terms, Factors & Coefficients | Parts of expression and multipliers | 5x → coefficient = 5 |
| Types of Expressions | Monomial, Binomial, Trinomial, Polynomial | x, x+3, x²+2x+1 |
• Variable = unknown value
• Coefficient = number part attached to variable
• Term = separated by + or −
• Coefficient = number part attached to variable
• Term = separated by + or −
If no number is written before a variable, its coefficient is 1.
Operations on Algebraic Expressions
| Operation | Rule | Example |
|---|---|---|
| Add / Subtract | Combine like terms | 3x + 2x = 5x |
| Monomial × Monomial | Multiply coefficients & variables | 2x × 3x = 6x² |
| Monomial × Polynomial | Distributive law | 2x(3x+1) = 6x²+2x |
| Polynomial × Polynomial | Multiply each term | (x+2)(x+3) |
| Standard Identities | Special multiplication rules | (a+b)² = a² + 2ab + b² |
Important identities:
(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
(a+b)(a−b) = a² − b²
(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
(a+b)(a−b) = a² − b²
Only like terms can be added or subtracted.
Exponents and Powers
| Law | Rule | Example |
|---|---|---|
| aᵐ × aⁿ | Add powers | a²×a³ = a⁵ |
| aᵐ ÷ aⁿ | Subtract powers | a⁵ ÷ a² = a³ |
| (aᵐ)ⁿ | Multiply powers | (a²)³ = a⁶ |
| Negative exponent | Reciprocal form | a⁻² = 1/a² |
| Scientific notation | Large numbers form | 5,000,000 = 5×10⁶ |
Negative exponent means reciprocal of the base.
Use laws step by step to avoid mistakes.
Linear Equations
| Type | Form | Example |
|---|---|---|
| One Variable | ax + b = 0 | 2x + 3 = 7 |
| Two Variables | ax + by = c | 2x + y = 5 |
| Pair of Linear Equations | Two equations together | x+y=5, x−y=1 |
| Solution Methods | Substitution / Elimination / Graph | Find x & y |
Solution of linear equation = Value of variable satisfying equation.
Always simplify equation before solving.
Polynomials
| Concept | Description | Example |
|---|---|---|
| Polynomial | Expression with powers of variable | 2x²+3x+1 |
| Degree | Highest power of variable | x³ → degree 3 |
| Zeroes | Value making polynomial zero | p(x)=0 |
| Division Algorithm | Dividend = Divisor × Quotient + Remainder | p(x)=dq+r |
Degree decides type: linear, quadratic, cubic.
Graph of polynomial touches x-axis at its zeroes.
Factorization and Division
| Process | Explanation | Example |
|---|---|---|
| Factorization | Breaking expression into factors | x²−9 = (x−3)(x+3) |
| Division (Monomial) | Term by term division | 6x² ÷ 3x = 2x |
| Division (Polynomial) | Using long/synthetic division | (x²+3x+2) ÷ (x+1) |
Factorization simplifies equations and solving.
Check result by multiplying factors back.
Coordinate Geometry and Linear Graphs
| Term | Meaning | Example |
|---|---|---|
| Coordinate | Point on graph | (2,3) |
| X-axis / Y-axis | Reference lines | Origin (0,0) |
| Linear Graph | Straight line graph | y = 2x + 1 |
Horizontal line → y constant
Vertical line → x constant
Vertical line → x constant
Plot points carefully using scale.
Arithmetic Progressions (AP)
| Concept | Formula | Example |
|---|---|---|
| nth term | aₙ = a + (n−1)d | 2,4,6 → d=2 |
| Sum of n terms | Sₙ = n/2 [2a+(n−1)d] | S₅ of 2,4,6… |
Common difference (d) = difference between consecutive terms.
Identify a and d first before applying formulas.
Geometric Progressions (GP)
| Concept | Formula | Example |
|---|---|---|
| nth term | aₙ = a·rⁿ⁻¹ | 2,6,18 → r=3 |
| Ratio | Common multiplier | Each term × 3 |
In GP, terms increase or decrease by a fixed ratio.
If r > 1 → increasing GP
If 0<r<1 → decreasing GP
If 0<r<1 → decreasing GP