Algebra
Polynomials, Equations, Exponents and Algebraic Identities
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Overview
Algebra forms the backbone of mathematics at the upper primary level (Classes 6–8) and carries significant weight in GTET Paper-2. This topic bridges arithmetic and higher mathematics, teaching students to work with unknown quantities, generalise patterns, and solve real-world problems symbolically.
For GTET, you must demonstrate both content mastery and pedagogical understanding. Questions typically test your ability to simplify expressions, solve equations, apply identities correctly, and explain why certain methods work. The syllabus expects familiarity with polynomials up to degree 2, linear equations in one and two variables, laws of exponents, and standard algebraic identities used in factorisation and simplification.
Mastering this topic also helps in geometry (coordinate geometry uses algebraic expressions) and mensuration (formulas involve algebraic manipulation). Expect 4–6 direct questions plus applications embedded in other areas.
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Key Concepts
- **Variable vs Constant**: A variable (x, y) represents an unknown or changing quantity; a constant (5, –3) has a fixed value. Understanding this distinction is foundational for forming expressions.
- **Algebraic Expression**: A combination of variables, constants, and operations (e.g., 3x² + 2x – 5). Terms are separated by + or – signs.
- **Polynomial**: An algebraic expression where variables have only whole number exponents. Classified by degree: linear (degree 1), quadratic (degree 2), cubic (degree 3).
- **Coefficient and Degree**: The coefficient is the numerical factor of a term (in 7x³, coefficient is 7). Degree of a polynomial is the highest power of the variable.
- **Equation vs Expression**: An expression has no equals sign; an equation asserts equality between two expressions and can be solved.
- **Exponent (Index/Power)**: In aⁿ, the exponent n tells how many times base a is multiplied by itself. Exponent rules allow simplification of complex expressions.
- **Algebraic Identity**: An equation true for all values of the variables (e.g., (a + b)² = a² + 2ab + b²). Identities are tools for quick expansion and factorisation.
- **Like and Unlike Terms**: Like terms have identical variable parts (3x² and –5x² are like terms). Only like terms can be combined through addition or subtraction.
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Formulas / Key Facts
### Laws of Exponents (for any non-zero base and integer exponents)