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104 LearnersLast updated on September 10, 2025
Partial Fraction Decomposition Calculator
Calculators are reliable tools for solving simple mathematical problems and advanced calculations like trigonometry. Whether you’re cooking, tracking BMI, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about partial fraction decomposition calculators.
What is Partial Fraction Decomposition Calculator?
A partial fraction decomposition calculator is a tool used to break down complex rational expressions into simpler fractions that are easier to integrate or differentiate.
This calculator simplifies expressions by expressing them as a sum of fractions with simpler denominators, making complex algebraic calculations more manageable.
How to Use the Partial Fraction Decomposition Calculator?
Given below is a step-by-step process on how to use the calculator:
Step 1: Enter the rational expression: Input the complex fraction into the given field.
Step 2: Click on decompose: Click on the decompose button to perform the decomposition and get the result.
Step 3: View the result: The calculator will display the decomposed fractions instantly.
How to Decompose a Rational Expression?
To decompose a rational expression into partial fractions, the calculator uses a systematic approach based on the degrees of the polynomials.
If the degree of the numerator is higher than or equal to the denominator, polynomial long division is performed first.
For a proper rational expression, the partial fraction decomposition takes the form:
1. For linear factors in the denominator: A / (x-a)
2. For irreducible quadratic factors: Ax+B / (x2+bx+c)
Thus, the expression is rewritten as a sum of these simpler fractions.
Tips and Tricks for Using the Partial Fraction Decomposition Calculator
When using a partial fraction decomposition calculator, consider the following tips to make the process easier and avoid errors:
Ensure that the expression is a proper fraction. If not, use polynomial long division.
Identify repeated factors and handle them with care, considering their multiplicity.
Verify the factorization of the denominator to ensure accuracy.
Use the calculator to check your manual calculations to ensure precision.
Common Mistakes and How to Avoid Them When Using the Partial Fraction Decomposition Calculator
Using a calculator doesn't eliminate the possibility of errors. Mistakes can occur, especially in complex expressions.
Mistake 1
Ignoring repeated factors in the denominator
When dealing with repeated factors, include terms for each power up to the highest occurring.
For example, for (x-a)2, include A / (x-a) + B/ (x-a)2.
Mistake 2
Not simplifying the rational expression first
Always simplify the expression before decomposition. If the numerator is of equal or higher degree than the denominator, perform polynomial long division first.
Mistake 3
Incorrect factorization of the denominator
Ensure the denominator is factored correctly into linear and irreducible quadratic factors. Incorrect factorization leads to errors in the decomposition process.
Mistake 4
Forgetting to verify solutions
After decomposition, recombine the partial fractions and simplify to verify the solution matches the original expression.
Mistake 5
Assuming all calculators will handle complex expressions effortlessly
Some calculators may not handle very complex or specific expressions accurately. Double-check results manually or with another tool if necessary.
Partial Fraction Decomposition Calculator Examples
Problem 1
Decompose the expression \(\frac{3x+5}{x^2-3x+2}\).
First, factor the denominator: x2-3x+2 = (x-1)(x-2)
Using partial fraction decomposition: 3x+5 / (x-1)(x-2) = A / x-1 + B / x-2
Solve for A and B by equating coefficients or substituting values of x.
3x+5 = A(x-2) + B(x-1)
Solving gives: A = 2, B = 1.
Thus, 3x+5 / x2-3x+2 = 2 / x-1 + 1 / x-2.
Explanation
By factoring the denominator, we express the rational expression as a sum of two partial fractions and solve for A and B.
Problem 2
Decompose \(\frac{2x^2+3x+1}{x^3-2x^2+x}\).
First, factor the denominator: x3-2x2+x = x(x-1)(x-1)
Using partial fraction decomposition: 2x^2+3x+1 / x(x-1)2 = A / x + B / x-1 + C / (x-1)2
Solve for A, B, and C using the system of equations derived from equating coefficients.
Explanation
The denominator is factored into linear and repeated factors, allowing decomposition into three partial fractions.
Problem 3
Find the partial fraction decomposition of \(\frac{x^2+4}{x^2+x-2}\).
Factor the denominator: x2+x-2 = (x+2)(x-1)
Using partial fraction decomposition: x2+4 / (x+2)(x-1) = A / x+2 + B / x-1
Solve for A and B.
Explanation
By factoring the quadratic denominator, the expression is decomposed into two simpler fractions.
Problem 4
Decompose \(\frac{x^3+2x^2+x}{x^4-1}\).
Factor the denominator: x4-1 = (x2+1)(x-1)(x+1)
Using partial fraction decomposition: x3+2x2+x / (x2+1)(x-1)(x+1) = Ax+B / x2+1 + C / x-1 + D / x+1
Solve for A, B, C, and D.
Explanation
The expression is decomposed by factoring the denominator into irreducible and linear factors.
Problem 5
Perform the partial fraction decomposition of \(\frac{5x^2+3x+7}{x^3+4x}\).
Factor the denominator: x3+4x = x(x2+4)
Partial fraction decomposition: 5x2+3x+7 / x(x2+4) = A / x + Bx+C / x2+4
Solve for A, B, and C.
Explanation
The expression is decomposed into partial fractions using the linear and irreducible quadratic factors.
FAQs on Using the Partial Fraction Decomposition Calculator
1.How do you decompose a rational expression into partial fractions?
2.Can all rational expressions be decomposed into partial fractions?
3.Why is partial fraction decomposition useful?
4.How do I use a partial fraction decomposition calculator?
5.Is the partial fraction decomposition calculator accurate?
Glossary of Terms for the Partial Fraction Decomposition Calculator
- Partial Fraction Decomposition: A method of expressing a complex fraction as a sum of simpler fractions.
- Proper Fraction: A rational expression where the degree of the numerator is less than the degree of the denominator.
- Polynomial Long Division: A process to divide polynomials, used when the numerator's degree is higher than the denominator's.
- Irreducible Quadratic: A quadratic expression that cannot be factored over the real numbers.
- Coefficient: A constant term related to the variables in an expression, solved during decomposition.
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Seyed Ali Fathima S
About the Author
Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
Fun Fact
: She has songs for each table which helps her to remember the tables
