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Partial Fraction Decomposition Calculator

Q: 5. What are the steps involved in partial fraction decomposition?

The steps generally involve: 1. Factor the denominator into its irreducible factors (linear or quadratic). 2. Set up the partial fraction form based on the type of factors in the denominator (distinct linear, repeated linear, irreducible quadratic). 3. Find the unknown constants (A, B, etc.) by comparing coefficients or substituting values of x. 4. Rewrite the original rational function as the sum of the partial fractions.

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How to Use the Partial Fraction Decomposition Calculator for Maths Problems

Partial Fraction Decomposition Calculator – Free Online Tool with Formula, Steps & Examples

Partial Fraction Decomposition Calculator

What is Partial Fraction Decomposition?

Partial fraction decomposition is a mathematical process in which a complex rational function (a fraction with polynomials in numerator and denominator) is expressed as a sum of simpler fractions, called partial fractions. This process makes it far easier to integrate or simplify algebraic expressions, solve integrals, or work with Laplace transformations. For example, (2x+3)/(x²+x) can be split into two simple fractions, each with a linear denominator.

Formula or Logic Behind Partial Fraction Decomposition Calculator

The logic is to factor the denominator and then write the original rational function as a sum of fractions whose denominators are the factors. Here are the common formulae:

Distinct Linear Factors: (ax+b)/(x²-1) = A/(x-1) + B/(x+1)
Repeated Linear Factors: 1/(x-1)² = A/(x-1) + B/(x-1)²
Irreducible Quadratic Factors: 1/(x²+1) = (Ax+B)/(x²+1)
Solve for the constants (A, B, etc) by equating numerators after clearing denominators, then use substitution or compare coefficients.

Partial Fraction Decomposition Examples Table

Expression	Partial Fraction Decomposition
1/(x²-1)	1/2 · 1/(x-1) - 1/2 · 1/(x+1)
(2x+3)/(x²+x)	2/x + 1/(x+1)
1/(x+2)²	A/(x+2) + B/(x+2)² (solve for A, B)
(x+7)/(x²+3x+2)	6/(x+1) + 5/(x+2)
1/(x²+1)	x/(x²+1) + (1)/(x²+1) (see complex roots case)

Steps to Use the Partial Fraction Decomposition Calculator

Enter the numerator polynomial (e.g., x^2+3).
Enter the denominator polynomial (e.g., x^2-1).
Click on the 'Calculate' button.
Instantly view the partial fraction decomposition and a quick step outline.

Why Use Vedantu’s Partial Fraction Decomposition Calculator?

Vedantu's calculator is easy to use, mobile-friendly, and ideal for CBSE, ICSE, JEE, and college students. It offers instant calculations, helps visualize formulas in steps, and is created and maintained by India’s leading Maths educators — making it reliable for homework, practice, and exam preparation.

Real-life Applications of Partial Fraction Decomposition

Partial fraction decomposition is widely used in mathematics and engineering. It's essential for integrating complex rational expressions in calculus, solving Laplace and inverse Laplace transforms in electrical and systems engineering, analyzing signals and circuits, simplifying algebraic fractions in higher maths, and even in financial modeling or control theory. Students encounter it frequently in competitive exams such as JEE, NEET, and school-level boards.

Want to explore more? Review Algebra topics for deeper maths practice. Try related calculators like the HCF Calculator or brush up your core concepts with our Prime Numbers and Polynomial sections. If you're preparing for competitive exams, you might also like our JEE Advanced Maths resources.

Developed by certified Maths educators, based on Indian and global curriculum standards. Accuracy-verified, regularly reviewed, and trusted by millions of students.

FAQs on Partial Fraction Decomposition Calculator

1. What is partial fraction decomposition?

Partial fraction decomposition is a technique in algebra used to rewrite a rational function (a fraction with polynomials in the numerator and denominator) as a sum of simpler fractions. This simplification makes it easier to integrate the function or solve related problems in calculus and other fields. The process involves expressing the original fraction as a sum of partial fractions, each with a simpler denominator.

2. What is the formula for partial fraction decomposition?

There isn't one single formula, but rather a set of rules depending on the factors in the denominator. For distinct linear factors like (x-a)(x-b), the partial fraction decomposition takes the form A/(x-a) + B/(x-b). For repeated linear factors like (x-a)², the form becomes A/(x-a) + B/(x-a)². For irreducible quadratic factors like (x²+bx+c), the form is (Ax+B)/(x²+bx+c). The constants A, B, etc., are found using algebraic methods like comparing coefficients or substituting specific values of x.

3. How do you find the values of A and B in partial fraction decomposition?

After setting up the partial fraction decomposition with the appropriate form, you solve for the unknown constants (A, B, etc.) using two main methods: 1) Comparing coefficients: Expanding the decomposed fractions and equating the coefficients of corresponding powers of x in the numerator to those in the original rational function. This leads to a system of linear equations that can be solved. 2) Substituting specific values of x: Choosing specific values of x (often the roots of the denominator) to simplify the equations and solve for the constants more directly.

4. How to decompose a rational function with repeated linear factors?

When the denominator has repeated linear factors, for example (x-a)², the partial fraction decomposition will include terms for each power of the repeated factor. For (x-a)², the decomposition would look like A/(x-a) + B/(x-a)². You'll need to solve for both A and B using the methods described above (comparing coefficients or substituting values).

5. What are the steps involved in partial fraction decomposition?

The steps generally involve: 1. Factor the denominator into its irreducible factors (linear or quadratic). 2. Set up the partial fraction form based on the type of factors in the denominator (distinct linear, repeated linear, irreducible quadratic). 3. Find the unknown constants (A, B, etc.) by comparing coefficients or substituting values of x. 4. Rewrite the original rational function as the sum of the partial fractions.

6. What is the difference between partial fraction and partial fraction decomposition?

They are essentially the same thing. 'Partial fraction' refers to the individual simpler fractions in the decomposed form, while 'partial fraction decomposition' is the process of breaking down a complex rational function into these simpler partial fractions.

7. What are some real-world applications of partial fraction decomposition?

Partial fraction decomposition is crucial in various areas, including: 1. Calculus: Evaluating integrals of rational functions. 2. Differential equations: Solving certain types of differential equations. 3. Signal processing: Analyzing and manipulating signals. 4. Control systems engineering: Designing and analyzing control systems. 5. Laplace transforms: Solving problems involving Laplace transforms.

8. How do you handle irreducible quadratic factors in partial fraction decomposition?

For irreducible quadratic factors (those that cannot be factored into linear terms with real coefficients), such as (x²+1), the corresponding partial fraction term will be of the form (Ax+B)/(x²+1). You'll then use the methods of comparing coefficients or substituting values to solve for A and B.

9. Can partial fraction decomposition be used for any rational function?

Yes, but the process might be more complex for higher-degree polynomials. The key is that the degree of the numerator must be strictly less than the degree of the denominator. If it's not, you'll first need to perform polynomial long division to obtain a quotient and a remainder, and then apply partial fraction decomposition to the remaining fraction.

10. What are some common mistakes to avoid when doing partial fraction decomposition?

Common errors include: 1. Incorrectly factoring the denominator. 2. Using an inappropriate form for the partial fractions based on the factors of the denominator. 3. Making errors in solving for the constants (A, B, etc.) through comparing coefficients or substitution. 4. Forgetting to perform polynomial long division if the degree of the numerator is greater than or equal to the degree of the denominator.

11. How do I use a partial fraction decomposition calculator?

Most calculators require you to input the numerator and denominator of your rational function separately. Then, simply click the 'Calculate' or similar button. The calculator will provide the decomposed partial fractions along with the steps followed to obtain the solution. Make sure you understand the steps shown, rather than relying solely on the calculator for every problem.