Algebra

Algebraic Expressions, Linear Equations, Identities & Factorisation

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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.

In the exam, expect 2–4 questions on algebra covering: forming and simplifying algebraic expressions, solving linear equations in one or two variables, applying algebraic identities, and factorising polynomials. 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.

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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).

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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) |