Polytechnic Semester 1 – Mathematics 1 (1004N)
Unit – Partial Fractions (à¤ागात्मक à¤िन्न)
Welcome students to D2D Pathshala. This page provides complete, exam-oriented notes of Partial Fractions for Diploma / Polytechnic 1st Semester Mathematics as per BTER syllabus. These notes are easy to understand, easy to remember, and very useful for exams.
1. What is Partial Fraction? (परिà¤ाषा)
A Partial Fraction is a technique used to break a complex rational expression into a sum of simpler fractions.
Rational Expression:
A fraction in which both numerator and denominator are polynomials is called a rational expression.
Example:
(5x + 7) / (x² − x − 2)
Using partial fractions, we convert this into simple fractions which are easy to solve, integrate, and understand.
Why Partial Fractions are important?
- Used in integration
- Simplifies algebraic expressions
- Very important for exams (2M, 5M, 10M questions)
📺 Video: Introduction to Partial Fractions
2. Types of Partial Fractions
Partial Fractions are classified based on the nature of denominator.
Type-1: Distinct Linear Factors
Form:
1 / [(x − a)(x − b)] = A/(x − a) + B/(x − b)
Example:
1 / [(x − 2)(x + 3)]
Memory Trick: 👉 One factor → One constant
Type-2: Repeated Linear Factors
Form:
1 / (x − a)² = A/(x − a) + B/(x − a)²
Memory Trick: 👉 Power increases → terms increase
Type-3: Linear × Quadratic Factor
Form:
(Ax + B) / (x² + px + q)
Note: Numerator must be linear when denominator is quadratic.
📺 Video: Types of Partial Fractions
3. Method of Solving Partial Fractions
Step-by-Step Procedure:
- Factorize the denominator completely
- Assume partial fractions according to type
- Multiply both sides by the denominator
- Compare coefficients or substitute values
- Find constants and write final answer
📌 Important Rule
If degree of numerator ≥ degree of denominator, first perform division.
📺 Video: Formula & Method
4. Solved Examples (Exam Oriented)
Example 1
Resolve into partial fractions:
(5x + 7) / [(x − 1)(x + 2)]
Solution:
Let:
(5x + 7)/[(x − 1)(x + 2)] = A/(x − 1) + B/(x + 2)
5x + 7 = A(x + 2) + B(x − 1)
Comparing coefficients:
A + B = 5 2A − B = 7
Solving: A = 4, B = 1
Final Answer:
4/(x − 1) + 1/(x + 2)
Example 2 (Repeated Factor)
(x + 3)/(x − 2)²
(x + 3)/(x − 2)² = A/(x − 2) + B/(x − 2)²
x + 3 = A(x − 2) + B
Comparing: A = 1, B = 5
Answer:
1/(x − 2) + 5/(x − 2)²
📺 Video: Solved Examples
5. Important Exam Questions
Q1. Resolve (2x + 5)/(x² + 3x)
Q2. Decompose (3x² + 7x + 2)/[(x + 1)(x² + 1)]
Q3. Express (4x − 1)/[(2x − 3)(x + 2)] as partial fractions
📺 Video: Practice & Exercise Questions
6. Extra Questions & Revision
✅ These notes are specially prepared for BTER Polytechnic exams. 📘 Learn concepts → Watch video → Practice questions → Score good marks.
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