Algebra
Algebraic Expressions, Linear Equations, Identities & Factorisation
Overview
Algebra forms the bridge between arithmetic and higher mathematics. For UPTET Paper I (Classes 1–5) and Paper II (Classes 6–8), algebra questions test your ability to manipulate expressions, solve equations, and apply standard identities—skills you will also need to teach effectively in the classroom.
Mastery here also strengthens your problem-solving in mensuration and data-handling topics where algebraic manipulation is often required.
The pedagogical section may ask how to introduce variables to young learners or how to correct common student errors in sign handling and factorisation. Hence, conceptual clarity is as important as computational speed.
Key Concepts
- Variable & Constant: A variable (x, y, a) represents an unknown quantity that can change; a constant (3, −7, π) has a fixed value.
- Algebraic Expression: A combination of variables, constants and operations (e.g., 3x² + 2x − 5). It does NOT have an equality sign.
- Equation: An expression set equal to another expression or a value (e.g., 2x + 3 = 11). Solving means finding the value of the variable that makes both sides equal.
- Like & Unlike Terms: Terms with identical variable parts (3xy and −5xy) are like terms and can be combined; unlike terms (3xy and 3x²) cannot.
- Degree of a Polynomial: The highest power of the variable present (e.g., degree of 4x³ − x + 7 is 3).
- Linear Equation: An equation where the highest power of every variable is 1. In one variable: ax + b = 0. In two variables: ax + by + c = 0.
- Identity: An equation true for ALL values of the variable(s), e.g., (a + b)² = a² + 2ab + b².
- Factorisation: Writing an expression as a product of its factors (reverse of expansion).
Formulas / Key Facts
Standard Algebraic Identities (must memorise)
| Identity | Expanded Form |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| a² − b² | (a + b)(a − b) |
| (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca |
| (x + a)(x + b) | x² + (a + b)x + ab |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
| (a + b)³ | a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) |
| (a − b)³ | a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a − b) |
Solving Linear Equations
- One variable: Isolate variable → ax + b = c ⟹ x = (c − b)/a
- Two variables (simultaneous): Use substitution or elimination; graphically, the solution is the intersection point of two lines.
Factorisation Methods
- Taking out common factor: 6x² + 9x = 3x(2x + 3)
- Grouping: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
- Using identities: x² − 16 = (x + 4)(x − 4)
- Splitting middle term: x² + 5x + 6 = (x + 2)(x + 3)
Worked Examples
Example 1 — Simplify and find value
Problem: Simplify 3(2x − 4) + 2(x + 5) and find its value when x = 3.
Solution: Step 1: Expand brackets → 6x − 12 + 2x + 10 Step 2: Combine like terms → 8x − 2 Step 3: Substitute x = 3 → 8(3) − 2 = 24 − 2 = 22
Example 2 — Solve a linear equation
Problem: Solve 5x − 3 = 2x + 9
Solution: Step 1: Bring variable terms to one side → 5x − 2x = 9 + 3 Step 2: Simplify → 3x = 12 Step 3: Divide → x = 4
Example 3 — Use identity to factorise
Problem: Factorise 4x² − 25
Solution: Recognise the form a² − b² where a = 2x, b = 5. Apply identity: (2x + 5)(2x − 5) Answer: (2x + 5)(2x − 5)
Example 4 — Splitting middle term
Problem: Factorise x² + 7x + 12
Solution: Find two numbers whose product = 12 and sum = 7 → 3 and 4. Split: x² + 3x + 4x + 12 = x(x + 3) + 4(x + 3) = (x + 3)(x + 4)
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Treating (a + b)² as a² + b² | Always include the middle term: a² + 2ab + b² |
| Sign errors when removing brackets after a minus sign: −(3 − x) = −3 − x | Distribute the negative to every term inside: −(3 − x) = −3 + x |
| Adding unlike terms: 2x + 3x² = 5x³ | Unlike terms stay separate: 2x + 3x² (cannot combine) |
| In simultaneous equations, forgetting to multiply the entire equation when eliminating | Multiply every term, including the constant, before adding/subtracting equations |
| Stopping factorisation too early: 2x(x + 3) + 6 left as is | Check if further grouping is possible: 2x(x + 3) + 6 does not simplify, but always verify |
Quick Reference
- (a + b)² = a² + 2ab + b² — never forget the 2ab.
- a² − b² = (a + b)(a − b) — difference of squares, very frequent in exams.
- To solve ax + b = c, isolate x: x = (c − b)/a.
- Factorising quadratics: find two numbers with product = constant term, sum = middle-coefficient.
- Like terms share the same variable AND same exponent; only like terms can be combined.
- An identity holds for all values; an equation holds for specific values of the variable.
Tip for pedagogy questions: Emphasise using concrete examples (matchsticks, tiles) to introduce variables, and encourage students to verify identities by substituting simple numbers before memorising formulas.