Class 9 Maths Exercise 4.8: Factorization & Algebra

Class 9 Mathematics Exercise 4.8 – Factorization and Algebraic Manipulation

Exercise 4.8 is the final and most important practice set of Chapter 4 in Class 9 Mathematics. It combines all previously learned factorization techniques and applies them to complex algebraic expressions and real-world word problems. This exercise is designed to test conceptual clarity, pattern recognition, and step-by-step algebraic manipulation skills.

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Concept Overview

Factorization is the process of expressing an algebraic expression as a product of its factors. In Exercise 4.8, students apply multiple identities and methods simultaneously instead of using a single technique. This improves logical thinking and strengthens algebraic fluency.

The key idea is to identify patterns first, then choose the correct method such as common factor, algebraic identities, or grouping.

Key Factorization Techniques Used

• Common factor method for simplifying expressions
• Difference of squares: a² − b² = (a + b)(a − b)
• Perfect square identities
• Grouping method for four-term expressions
• Cubic identities in volume-based problems

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Worked Example 1: Algebraic Area Problem

A rectangular region has an area expressed as x² − 2x − 3. To find its dimensions, we factorize the expression.

Step 1: Split the middle term to match factors of −3 and −2 

Step 2: Rewrite expression as x² − 3x + x − 3 

Step 3: Group terms 

Step 4: Factor common terms Result: (x − 3)(x + 1)

This gives the dimensions of the rectangle as (x − 3) and (x + 1). Such problems frequently appear in exam word problems involving area and cost.

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Worked Example 2: Square and Perimeter Application

Consider a square with area 25x² − 30x + 9. We recognize this as a perfect square trinomial.

Step 1: Identify square roots of first and last term 

Step 2: Check middle term consistency Result: (5x − 3)²

Now, side length = 5x − 3 Perimeter = 4(5x − 3) = 20x − 12

This shows how factorization directly connects to geometry problems.

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Worked Example 3: Cubic Expression Interpretation

In more advanced problems, expressions like 125x³ − 150x² + 60x − 8 appear. These match cubic expansion patterns.

Step 1: Recognize structure of (a − b)³ 

Step 2: Compare coefficients 

Step 3: Identify a = 5x and b = 2 Result: (5x − 2)³

Such problems help students connect algebra with real-world applications like volume calculations.

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How to Approach Exam Questions

1. Always check for a common factor first.
2. Identify whether the expression fits a known identity.
3. Break long expressions into smaller grouped parts.
4. Verify your answer by expansion.
5. Practice mixed exercises instead of single-method problems.

Mastering these steps ensures strong performance in board exams, especially in structured and application-based questions.

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Conclusion

Exercise 4.8 is not just practice; it is a complete revision of the entire factorization chapter. Students who can solve this exercise independently have strong command over algebraic manipulation. Regular practice builds speed, accuracy, and confidence for exam success.