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)

| Rule | Formula | |------|---------| | Product Rule | aᵐ × aⁿ = aᵐ⁺ⁿ | | Quotient Rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | | Power of a Power | (aᵐ)ⁿ = aᵐⁿ | | Power of a Product | (ab)ⁿ = aⁿbⁿ | | Power of a Quotient | (a/b)ⁿ = aⁿ/bⁿ | | Zero Exponent | a⁰ = 1 | | Negative Exponent | a⁻ⁿ = 1/aⁿ |

### Standard Algebraic Identities (Class 8 level)

1. (a + b)² = a² + 2ab + b² 2. (a – b)² = a² – 2ab + b² 3. (a + b)(a – b) = a² – b² 4. (x + a)(x + b) = x² + (a + b)x + ab 5. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca 6. (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b) 7. (a – b)³ = a³ – 3a²b + 3ab² – b³ = a³ – b³ – 3ab(a – b) 8. a³ + b³ = (a + b)(a² – ab + b²) 9. a³ – b³ = (a – b)(a² + ab + b²)

### Linear Equations

• **One variable**: ax + b = 0 → x = –b/a
• **Two variables**: ax + by = c (represents a straight line; needs two equations for unique solution)

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Worked Examples

### Example 1: Simplify using exponent laws **Problem**: Simplify (2³ × 2⁵) ÷ 2⁴

**Solution**:

• Step 1: Apply product rule in numerator: 2³ × 2⁵ = 2³⁺⁵ = 2⁸
• Step 2: Apply quotient rule: 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴
• Step 3: Evaluate: 2⁴ = 16

**Answer**: 16

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### Example 2: Expand using identity **Problem**: Expand (3x + 5)²

**Solution**:

• Use identity (a + b)² = a² + 2ab + b²
• Here a = 3x, b = 5
• (3x)² + 2(3x)(5) + 5²
• = 9x² + 30x + 25

**Answer**: 9x² + 30x + 25

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### Example 3: Factorise using identity **Problem**: Factorise 49y² – 64

**Solution**:

• Recognise as difference of squares: a² – b² = (a + b)(a – b)
• 49y² = (7y)² and 64 = 8²
• = (7y + 8)(7y – 8)

**Answer**: (7y + 8)(7y – 8)

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### Example 4: Solve linear equation **Problem**: Solve 3(x – 2) + 7 = 2x + 5

**Solution**:

• Step 1: Expand: 3x – 6 + 7 = 2x + 5
• Step 2: Simplify left side: 3x + 1 = 2x + 5
• Step 3: Transpose: 3x – 2x = 5 – 1
• Step 4: Solve: x = 4

**Verification**: LHS = 3(4–2) + 7 = 6 + 7 = 13; RHS = 2(4) + 5 = 13 ✓

**Answer**: x = 4

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Common Mistakes

1. **Confusing exponent addition with multiplication**

• Wrong: 2³ × 2⁴ = 2¹² (multiplying exponents)
• Correct: 2³ × 2⁴ = 2⁷ (add exponents when bases are same and multiplied)

2. **Forgetting the middle term in square identities**

• Wrong: (a + b)² = a² + b²
• Correct: (a + b)² = a² + 2ab + b² (the 2ab term is essential)

3. **Sign errors in (a – b)² expansion**

• Wrong: (a – b)² = a² – 2ab – b²
• Correct: (a – b)² = a² – 2ab + b² (last term is always positive)

4. **Applying a² – b² identity to a² + b²**

• Wrong: a² + b² = (a + b)(a – b)
• Correct: a² + b² cannot be factorised using real numbers; only a² – b² uses this identity

5. **Transposition sign errors in equations**

• Wrong: If 3x = 2x + 5, then 3x + 2x = 5
• Correct: 3x – 2x = 5 (when moving terms, change the sign)

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Quick Reference

• **Polynomial degree** = highest exponent of the variable in the expression

• **aᵐ × aⁿ = aᵐ⁺ⁿ** (same base: add powers) | **aᵐ ÷ aⁿ = aᵐ⁻ⁿ** (same base: subtract powers)

• **(a + b)² = a² + 2ab + b²** — never forget the middle term

• **(a + b)(a – b) = a² – b²** — difference of squares; quick factorisation tool

• **To solve equations**: simplify → collect variable terms on one side → solve → verify

• **Like terms only**: 5x² + 3x² = 8x², but 5x² + 3x cannot be combined