Algebra

Linear equations, expressions, identities and factorisation

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Overview

Algebra forms the bridge between arithmetic and higher mathematics, introducing students to the power of symbols and generalisation. For PSTET Paper II, algebra questions test both your conceptual understanding and your ability to teach these abstract ideas to Classes VI-VIII students. Expect questions on simplifying expressions, solving linear equations, applying standard identities, and factorising polynomials.

This topic carries significant weight because it underpins problem-solving across mathematics. Examiners often frame questions as word problems requiring equation formation, or ask you to identify errors in student work—testing pedagogical awareness alongside content mastery. A strong grip on identities and factorisation techniques is essential, as these appear repeatedly in both content and pedagogy sections.

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Key Concepts

• **Algebraic Expression**: A combination of variables, constants, and operations (e.g., 3x + 5y − 7). Unlike equations, expressions have no equality sign.

• **Terms, Coefficients, and Constants**: In 4x² − 3x + 2, there are three terms; 4 and −3 are coefficients; 2 is the constant term.

• **Like and Unlike Terms**: Terms with identical variable parts (e.g., 5xy and −2xy) are like terms and can be combined; unlike terms cannot.

• **Linear Equation in One Variable**: An equation of the form ax + b = 0 where a ≠ 0. The solution is x = −b/a.

• **Linear Equation in Two Variables**: Form ax + by + c = 0. Represents a straight line; infinite solutions exist as ordered pairs (x, y).

• **Algebraic Identity**: An equality true for all values of variables. Identities are used for expansion and factorisation—not to be confused with equations that hold only for specific values.

• **Factorisation**: Expressing a polynomial as a product of its factors. Reverse of expansion.

• **Polynomial Degree**: The highest power of the variable. Degree 1 = linear, Degree 2 = quadratic.

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Formulas / Key Facts

### Standard Algebraic Identities (Class VIII)

| Identity | Expanded Form | |----------|---------------| | (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 | | (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca |

### Solving Linear Equations

• **Transposition Rule**: When a term moves across the equals sign, its sign changes.