Partial Fraction
2026-02-28 10:39 Diff
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Last updated on October 21, 2025

Breaking a complex rational expression into a sum of simpler fractions is called partial fractions. It is especially used in integration, Laplace transforms, and to solve differential equations. In this article, we will learn more about the partial fractions.

What is Partial Fraction?

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Breaking down complicated fractions into smaller parts is known as partial fraction. These smaller parts are easier to work with when we add, subtract, or integrate them. Like fractions, partial fractions also have a numerator and a denominator. When the denominator has multiple factors, we can split the fraction into a sum of smaller ones. 

What are the Formulas for Partial Fractions?

We use specific formulas to solve partial fractions based on the type of denominator. These formulas tell us to choose a kind of numerator that can be used for each type of denominator. These formulas help us rewrite difficult fractions, which makes integration and simplification easier. The formulas for partial expressions are mentioned below:

Type Form of Rational Expression Partial Fraction Decomposition

Non-repeated linear factor

\(\frac{P(x)}{(ax + b)^n} \) \(\frac{A}{ax + b} \)

Repeated linear factor

\(\frac{P(x)}{(ax + b)^n} \) \(\frac{A_1}{(ax + b)} + \frac{A_2}{(ax + b)^2} + \cdots + \frac{A_n}{(ax + b)^n} \)

Non-repeated quadratic factor

\(\frac{P(x)}{ax^2 + bx + c} \) \(\frac{Ax + B}{ax^2 + bx + c} \)

Repeated quadratic factor

\(\frac{P(x)}{(ax^2 + bx + c)^n} \) \(\frac{A_1x + B_1}{(ax^2 + bx + c)} + \cdots + \frac{A_nx + B_n}{(ax^2 + bx + c)^n} \)

How to Decompose Partial Fractions?

Breaking a complex fraction into simpler fractions is called partial fraction decomposition. We can learn easily about the partial fraction decomposition using the following steps:

Step 1: Write the given fraction as a sum of simpler fractions. For example, \(\frac{7x + 4}{(x + 1)(x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2} \).

Step 2: To eliminate the denominator, multiply both sides by \((x + 1)(x + 2) \). Therefore, the equation will become \(7x + 4 = A(x + 2) + B(x + 1) \).
 

Step 3: Now open the brackets and solve the equation.

\(7x + 4 = A(x + 2) + B(x + 1) \)

\(7x + 4 = Ax + 2A + Bx + B \)

\(7x + 4 = (A + B)x + (2A + B) \)

Coefficients of x:\( A + B = 7\)

Constant term: \(2A + B = 4\).

Step 4: Solve for A and B

\((1) A + B = 7\)

\((2) 2A + B = 4\)

Subtract equation (1)  from equation (2):

\((2A + B = 4) - (A + B = 7) = 4 - 7\)

When subtracting both equations, the B term gets cancelled, and we will get the value of A as: 

A = -3

Substitute A = -3 in the first equation,

A + B = 7

-3 + B = 7

B = 7 + 3

B = 10

Therefore, the final answer becomes: \(\frac{7x + 4}{(x + 1)(x + 2)} = \frac{-3}{x + 1} + \frac{10}{x + 2} \)

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Partial Fractions of Improper Fractions

When the numerator in a fraction is larger than the denominator, then it is called an improper fraction. Convert the improper fraction to a proper one before breaking the fraction into smaller parts. To make the improper fraction into a proper fraction, we use the long division method. Follow the steps given below for the improper fraction of partial fractions.

Step 1: Do the long division method. Divide the numerator by the denominator.

Step 2: After long division, we write the numbers in the form: Quotient + Remainder / Divisor.

So, the improper fraction becomes a polynomial and a smaller proper fraction.

Partial Fractions in Integration

Integration using partial fractions involves breaking a fraction into smaller parts and integrating each. Follow the steps given below:

Step 1: Break the given fraction into smaller fractions,

.

Step 2: Find the values of A and B.
 

Step 3: Now the equation will become,

Step 4: We can integrate this easily using the formula:

Tips and Tricks to Master Partial Fraction

Learning partial fractions makes it easier to simplify complex rational expressions and solve integrals. These tips help improve accuracy and speed in decomposition.

  • Always factor the denominator completely before starting the decomposition.
     
  • Identify the type of factors, linear, repeated linear, or quadratic to choose the correct form for the partial fractions.
     
  • Use substitution or equating coefficients to solve for unknown constants efficiently.
     
  • Check your work by combining the partial fractions to ensure they equal the original expression.
     
  • Practice different forms of rational expressions regularly to recognize patterns and improve speed.

Common Mistakes and How To Avoid Them in Partial Fractions

Students make mistakes when learning partial fractions. Listed below are some of the common mistakes that they make, and the ways to avoid them, and to help them avoid those mistakes.

Real Life Applications of Partial Fractions

In real life, partial fractions are widely used in fields such as mathematics, engineering, and science. Listed below are some real-life applications where partial fractions are used.  

  • Control system engineering: Partial fractions are used to decompose complex transfer functions for analyzing system stability and designing controllers.
     
  • Signal processing: In digital and analog signal processing, partial fractions simplify Laplace and Z-transform expressions for filtering, modulation, and system analysis.
     
  • Differential equations in physics: They help solve higher-order differential equations in mechanics, electromagnetism, and quantum physics by breaking complex expressions into integrable parts.
     
  • Communication systems: Partial fractions are applied in analyzing and simplifying frequency response equations and Fourier transforms in telecommunications.
     
  • Applied mathematics and modeling: Advanced population dynamics, epidemic spread models, and chemical reaction kinetics often use partial fraction decomposition to solve complex rational expressions efficiently.

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Problem 1

Break into partial fractions: 5/((x + 1)(x + 2))

Okay, lets begin

Explanation

We can write the equation as:

Multiply on both sides to remove the denominators. 

Comparing both sides, there is no x coefficient, so it can be written as 0.

A + B = 0,

B = -A

Substitute the value of A with

A = 5

Put A = 5, in B = -A

B= -5

Therefore, the final values in

Well explained 👍

Problem 2

Break this into partial fractions: x + 5/(x + 1)2

Okay, lets begin

Explanation

The equation will become,

Multiply (x +1)2 on both sides,

Comparing both sides,
A = 1

Substitute A = 1 in A + B = 5

A + B = 5

1 + B = 5

B = 5 - 1

B = 4

Therefore, the answer becomes,

Well explained 👍

Problem 3

Break this into partial fractions: 7x + 5/(x + 1)(x + 2)

Okay, lets begin

Explanation

We can write the equation as,

Comparing both sides,

A + B = 7, 2A + B = 5

Substitute the value of A for

A + B = 7

-2 + B = 7

B = 7 + 2

B = 9

Therefore, the final answer becomes,

Well explained 👍

Problem 4

Break this into partial fractions: 2x + 3/(x + 1)2

Okay, lets begin

Explanation

The equation will become, 

Multiply both sides by (x + 1)2:

Compare both sides,

A = 2,

Substitute A = 2 in A + B = 3

A + B = 3

2 + B = 3

B = 3 – 2

B = 1

Well explained 👍

Problem 5

Break this into partial fractions: (5x + 9)/((x + 2)(x + 3))

Okay, lets begin

5x + 9/(x + 2)(x + 3) = -1/(x + 2) + 6/(x + 3)

Explanation

The equation is:

(5x + 9)/((x + 2)(x + 3)) = A/x + 2 + B/x + 3

Multiply both the sides by (x +2)(x + 3),

5x + 9 = A(x + 3) + B(x + 2)

5x + 9 = Ax + 3A + Bx + 2B

5x + 9 = (Ax + Bx) + (3A + 2B)

5x + 9 = (A + B)x + (3A + 2B)

Comparing both sides,

A + B = 5, 3A + 2B = 9

Simplify the equation,

A + B = 5

B = 5 - A

Substitute B = 5 - A to 3A + 2B = 9

A + 10 = 9

A = 9 - 10

A = -1

Substitute A = -1 to A + B = 5

A + B = 5

-1 + B = 5

B = 5 + 1

B = 6

Therefore, the equation will become

Well explained 👍

FAQs of Partial Fractions

1.What are partial fractions?

A complicated fraction that is divided into smaller parts is known as partial fractions. 

2.Why do we use partial fractions?

We use partial fractions to make the problem easier, solve integrals in calculus, and simplify functions in physics and engineering.

3.When can a fraction be split?

A fraction can be split only when the numerator is smaller than the denominator.

4.Where are partial fractions used in real life?

Partial fractions are used in calculus, engineering, physics, economics, computer science, etc.

5.Can partial fractions have more than three parts?

Yes, if the denominator has many factors, we will have many parts. 

6.How can I explain partial fractions to my child in simple terms?

You can compare it to dividing a pizza into smaller, manageable slices. A complicated fraction is split into simpler parts that are easier to work with.

7.How can I help my child practice partial fractions at home?

Start with simple examples like fractions with linear denominators, then gradually move to more complex expressions. Encourage them to verify their results by combining the fractions back together.

8.What are partial fractions and why are they important?

Partial fractions are a way to break down complex rational expressions into simpler fractions. Learning this helps children solve advanced algebra and calculus problems more easily.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.