Algebra — Study Notes for Bihar TET Paper II

Overview

Algebra forms the backbone of upper-primary mathematics and carries significant weight in the Bihar TET Paper II Mathematics section. This topic tests your ability to manipulate symbols, simplify expressions, and solve equations—skills that directly connect arithmetic to higher mathematics.

For the exam, you must demonstrate two competencies: (1) fluency in algebraic operations and standard identities, and (2) the ability to set up and solve linear equations from word problems. Questions typically range from straightforward simplification to application-based problems involving age, money, or geometric relationships.

Mastering algebra is also essential for teaching Classes 6–8, where students transition from concrete numbers to abstract symbols. A teacher who understands common student errors in algebra can design better lessons and assessments.

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Key Concepts

• **Variable and Constant**: A variable (like x, y) represents an unknown or changing quantity; a constant (like 5, −3) has a fixed value.

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

• **Terms, Coefficients, and Like Terms**: In 4x² + 3x − 2, there are three terms; 4 is the coefficient of x²; like terms share the same variable and power (e.g., 5x and −2x).

• **Polynomial Classification**: Monomial (1 term), Binomial (2 terms), Trinomial (3 terms). Degree = highest power of the variable.

• **Algebraic Identity**: An equation true for all values of the variable—distinct from an equation which is true only for specific values.

• **Linear Equation in One Variable**: Has the form ax + b = 0, where a ≠ 0. The solution is a single value of x.

• **Linear Equation in Two Variables**: Has the form ax + by + c = 0. Graphically represents a straight line; needs two equations to find a unique solution.

• **Transposition Rule**: When moving a term across the equality sign, change its sign (+ becomes −, × becomes ÷).

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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 |

### Useful Facts

• **Degree of a polynomial**: Highest exponent of the variable (e.g., degree of 5x³ − 2x + 1 is 3).