Partial Fractions

In algebra, partial fraction is a powerful tool to simplify complex rational expressions that are often difficult to solve or integrate directly. By breaking down these expressions into simpler parts, we can remove the complexity. This technique is especially useful in calculus, integration, and solving algebraic equations. Let’s learn in detail about what a partial fraction is, important formulas, and how to solve them with sample problems.

Table of Content

• Partial Fractions Definition
• Formulas of Partial Fractions
• Type of Partial Fractions
• Importance of Partial Fractions
• Frequently Asked Questions on Partial Fractions
  
  

Partial Fractions Definition

When an algebraic expression, especially a rational expression, is split into the sum of two or more simpler rational expressions, each with a simpler denominator, it is called a partial fraction. In simple terms, the process of breaking one complicated fraction into several simpler ones is called a partial fraction. 

In basic terms, if you have a fraction like 𝑃(𝑥)/𝑄(𝑥) where both P(x) and Q(x) are polynomials, you can sometimes rewrite it as a sum of simpler fractions like (x−a)/A+(x−b)/B.​

Formulas of Partial Fractions

Some of the partial fraction formulas are given below to help you understand how to decompose a rational expression into partial fractions. These are common types of partial fractions, which are used to solve problems.

Common Partial Fraction Decomposition Forms

Type of Partial Fractions

Based on the factors of the denominator of a rational expression, we can classify partial fractions into the following types:

I. Distinct Linear Factors:

1/[(x+2)(x−3)] = A/(x+2) ​+ B/(x−3)​

II. Repeated Linear Factors:

1/(x−1)²=A/(x−1)+B(x−1)²

III. Irreducible Quadratic Factors

1/(x² + 1) = (Ax + B)/(x² + 1)

Why Use Partial Fractions?

Sometimes algebraic fractions are complex, and solving them requires splitting them into smaller parts; by doing so we can:

• Integrate functions easily in calculus.

• Solve differential equations.

• Simplify and transform higher-level engineering concepts.

• Make algebraic manipulation easier when solving rational equations.
  
  

Importance of Partial Fractions

Here’s why partial fractions matter in both theory and practical applications:

1. Simplifying Complex Expressions

Long or complicated fractions can be broken into easier parts, making them more manageable for solving or graphing.

2. Integration in Calculus

Some functions can’t be integrated in their original form. Decomposing them into partial fractions allows us to use basic integration rules.

3. Solving Rational Equations

Algebraic equations involving rational expressions are easier to solve once they are split into partial fractions.

4. Helps Build Algebraic Intuition

Understanding partial fractions improves your grasp of polynomials, factorization, and how functions behave.

Frequently Asked Questions on Partial Fractions

1: What is meant by partial fractions?

Answer: In mathematics, the partial fraction is defined as the process of decomposition of a fraction into the simplest form of the fraction.

2: Write down the procedure for partial fraction decomposition.

Answer: The procedure for the partial fraction decomposition is as follows:

i) In a given rational expression, factor the denominator into the linear factors.

ii) For each factor obtained, write down the partial fraction with variables in the numerator, say x and y. 

iii) To remove the fraction, multiply the whole equation by the denominator factor. Now, solve for the constants x and y.

iv) Substitute the constant values in the numerators of the partial fraction, and you will get the solution.

3. When is partial fraction decomposition used?

Answer: Partial fraction decomposition is used when the degree of the numerator is less than the degree of the denominator in a rational expression.

4: What if the numerator's degree is higher?

Answer: If the numerator's degree is higher, first use the long division method to divide the polynomials and then decompose the remainder using partial fractions.