📘 Exercise 5.2 | Partial Fractions | New First Year Math 2025 | Complete Solution by Sir Khawar Mehmood
Welcome to Math Universe Online! In this comprehensive and student-friendly video, Sir Khawar Mehmood explains and solves the complete Exercise 5.2 from the New First Year Mathematics Book 2025. This chapter deals with the very important algebraic concept of Partial Fractions.
This video is designed especially for students of FSc Pre-Engineering, I.CS, I.Com, and FA (Mathematics). It covers all the concepts, methods, and questions required for a strong understanding of partial fractions. Sir Khawar uses simple language and a step-by-step approach to make every topic easy to understand, even for beginners.
📖 What You Will Learn in This Lecture:
🔹 Introduction to Partial Fractions
The video begins with a basic and clear explanation of what partial fractions are. Partial fractions are a method to break down a complex rational expression into smaller, simpler parts. These smaller parts are easier to work with, especially when solving algebraic problems or integrating in calculus.
For example, when a fraction has a complicated denominator with brackets, we can separate it into smaller fractions using the method of partial fractions. This process is called decomposition.
🔹 Definition of Equations, Conditional Equations, and Identities
Sir Khawar also discusses three basic but important terms:
Equation: A statement that shows two expressions are equal.
Conditional Equation: An equation that is true only for certain values.
Identity: An equation that is true for all values of the variable.
Understanding identities is especially important when checking your solution after breaking a fraction into parts. By comparing both sides of the equation, you can confirm whether your values are correct.
🔹 Types of Rational Algebraic Expressions
A rational algebraic expression is one in which one polynomial is divided by another. These expressions are divided into two types:
1. Proper Rational Expression: The degree (or power) of the numerator is less than the degree of the denominator.
2. Improper Rational Expression: The degree of the numerator is equal to or greater than the degree of the denominator.
In the case of an improper expression, you must first use long division to simplify it before you can apply the method of partial fractions. Sir Khawar demonstrates this process clearly in the video.
🔹 Types of Partial Fractions Covered in Exercise 5.2
In Exercise 5.1, students learned how to deal with linear factors. In this exercise, you will go one step further and learn two new and slightly advanced types of partial fractions:
🟦 1. Non-Repeated Irreducible Quadratic Factors
These are factors that cannot be split or simplified into smaller linear terms using real numbers. For such cases, the partial fraction is written with a linear term in the numerator (for example, Ax + B) over the quadratic term in the denominator.
Sir Khawar explains how to:
Set up the equation correctly
Multiply both sides by the denominator
Expand the right-hand side
Collect like terms
Compare coefficients and solve for unknown values
He provides multiple examples to make this method very clear and easy to follow.
🟦 2. Repeated Irreducible Quadratic Factors
This is the case when a quadratic factor that cannot be simplified appears more than once in the denominator. For this type, we write two separate fractions, one for the factor and one for its square, each with a linear term in the numerator.
This part of the exercise is more challenging, but Sir Khawar explains it in a slow, clear, and detailed manner. He walks through each step of solving for the unknown values by comparing terms and solving the resulting system of equations.
✅ Full Solution of Exercise 5.2
Sir Khawar solves each and every question of Exercise 5.2 from the new first year math book. Some of the problems involve:
Splitting a rational expression into simpler fractions using non-repeated quadratic factors
Solving expressions where the quadratic factor is repeated
Using long division for improper fractions before decomposition
Applying the method of comparing coefficients to find unknown values
All questions are solved live on the board in the video with detailed explanation and useful tips. Nothing is skipped.
🌟 Key Highlights of This Lecture
Full coverage of Exercise 5.2 from Punjab Textbook Board 2025
Detailed explanation of all types of rational expressions
Slow-paced, clear Urdu explanation
Concept-based teaching suitable for both average and bright students
Solved examples following board exam patterns
Great for both classroom learning and self-study
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