Partial Fractions: Formulas, Decomposition & Examples

Partial Fractions are the simplest fractions formed when the complex rational expression is split into two or more simpler fractions. In Mathematics, fractions with algebraic expressions are complex to solve, thus, Partial Fractions are used to split these complex fractions into numerous simpler subfractions.

• Partial Fractions are defined as the fractions obtained after the decomposition of a rational expression. 
• An algebraic expression is split into a sum of two or more rational expressions with each part being referred to as a Partial Fraction.
• Partial Fractions are the reverse operation of the addition of rational expressions.
• Similar to fractions, partial fractions also have a numerator and denominator.
• The denominator is an algebraic expression that represents the decomposed part of a rational function.

Partial Fractions play an integral role in the concept of Integration, particularly, Integration by Partial Fractions. 

Read More: NCERT Solutions For Class 12 Mathematics Integrals

Key Terms: Partial Fractions, Rational Functions, Integration by Partial Fractions, Partial Fraction Decomposition, Algebraic Expressions

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What is Partial Fraction?

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Partial Fractions are the simpler fractions obtained after the decomposition of a complex algebraic fraction.

• The denominator of complex algebraic expressions is factorized to obtain the set of partial fractions.
• Partial Fractions also have a numerator and denominator just like general factions. 
• They are the opposite of the addition of rational expressions.
• Partial Fraction Decomposition is the method of splitting a rational expression into its smaller partial fractions.
• Partial Fractions are used in Integration by Partial Fractions.

Partial Fraction Example

Consider an algebraic fraction, (3x + 5) / (2x²-5x-3). The given expression can be split and written as: 

\(\frac{3x + 5}{2x^2-5x-3} = \frac{2}{x-3} - \frac{1}{2x+1}\)

Here .  

• \(\frac{3x + 5}{2x^2-5x-3}\) is the Rational Expression.
• \(\frac{2}{x-3}\) and \(\frac{1}{2x+1}\)are the Partial Fractions.

It must be noted that the Partial Fraction Decomposition is only applicable for proper rational expression, i.e. the degree of the numerator is less than the degree of the denominator. In the case of improper rational expression where the degree of the numerator is greater than the degree of the denominator, it needs to be converted into a proper rational expression using the polynomial long division method.

Integrals Detailed Video Explanation

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Partial Fraction Formulas

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Partial Fraction Formulas are the mathematical formulas that are used for the decomposition of rational expressions into partial fractions. Here are the common types of partial fractions used in Mathematics to solve different problems: 

Where, A, B, and C are Real Numbers.

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Partial Fractions of Rational Functions

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Rational Numbers are defined as those numbers that can be represented in the form of p/q, such that p and q are integers and q≠0.

• Rational Function is the ratio of two polynomial functions P(x) and Q(x), where P and Q are polynomials in x and Q(x)≠0. 
• A rational function is considered proper if the degree of P(x) is less than the degree of Q(x).
• If the degree of P(x) is more than the degree of Q(x), it is called an Improper Rational Function. 
• Improper Rational Functions can be converted to proper rational functions using the polynomial long division process.

Thus, if \(\frac{Px}{Qx}\)is improper, then it can be expressed as follows: 

\(\frac{P(x)}{Q(x)} = A + \frac{R(x)}{Q(x)}\)

Where A is a polynomial and R(x)/Q(x) refers to a Proper Rational Function. 

Integration of a function f(x) is given by F(x) and is expressed as follows: 

Here the above equation represents the integral of f(x) with respect to x. 

A few examples of partial fractions are as follows: 

• 4/[(x - 1)(x + 5)] = [A/(x - 1)] + [B/(x + 5)]
• (3x + 1)/[(2x - 1)(x + 2)²] = [A/(2x - 1)] + [B/(x + 2)] + [C/(x + 2)²]
• (2x - 3)/[(x - 2)(x² + 1)] = [A/(x - 2)] + [(Bx + C)/(x² + 1)]

In the above examples, A, B, and C are constants to be determined. 

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Partial Fraction Decomposition

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Partial Fraction Decomposition is the method in which the integrand is expressed as the sum of simpler rational functions. In order to integrate a rational function, it is reduced to a proper rational function. The stepwise process to decompose a fraction into partial fractions is mentioned below: 

• Step 1: Start with the proper rational expression to decompose the rational expressions into partial fractions.
• Step 2: Factorize the numerator and denominator to simplify the rational expression.
• Step 3: Split the rational expression using Partial Fraction Formula P/((ax + b)² = [A/(ax + b)] + [B/(ax + b)²]. 
• Step 4: Find the LCM of the factors of the denominators of partial fractions, and multiply both sides of the equation with the LCM.
• Step 5: Simplify and find the values of A and B through the comparison of coefficients of like terms on both sides.
• Step 6: Lastly, substitute the values of A and B on the right side of the equation to get the partial fractions. 

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Partial Fraction of Improper Fraction

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Improper Fraction is an algebraic fraction in which the degree of the numerator is greater than or equal to that of the denominator. The degree is the highest power of a polynomial.

• In order to an improper fraction into a partial fraction, long division needs to be performed.
• Long Divison gives a whole number and a proper fraction.
• The whole number forms the quotient in the long division, the remainder is the numerator, and the denominator is the divisor.
• The long division would result in the form Quotient + Remainder/Divisor. 

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Partial Fraction in Integration

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Integration by Partial Fractions is a method of integration in which partial fractions are used. In integration by partial fractions, the partial rational fraction is decomposed into a sum of simpler rational fractions through the process of Partial Fraction Decomposition.

Here is a detailed example of Integration using Partial Fractions.

Example 1: Evaluate ∫ [(x² + 1) / (x² – 5x + 6)] dx. 

Solution: Given expression is ∫ [(x² + 1) / (x² – 5x + 6)] dx.

The rational function is not a proper rational function. Thus, we need to decompose the same. 

(x² + 1) / (x² – 5x + 6) = 1 + (5x – 5) / (x² – 5x + 6) = 1 + (5x – 5) / (x – 2) (x – 3)

On observing the second half of the equation, it can be stated that,

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

On further simplification, 

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

Now, we get A + B = 5 and 3A + 2B = 5 by comparing the coefficients of the x term and constants. On solving the equations, we get 

• A = – 5
• B = 10 

Substituting the values, we get

(x² + 1) / (x² – 5x + 6) = 1 – 5 / (x – 2) + 10 / (x – 3)

Thus, ∫ [(x² + 1) / (x² – 5x + 6)] dx = ∫ dx – 5 ∫ 1 / (x – 2) + 10 ∫ 1 / (x – 3) = x – 5log |x – 2| + 10log |x – 3| + C.

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Partial Fractions Solved Examples

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Given below are some solved examples on Partial Fractions to understand the concept better: 

Example: Determine the Partial Fraction Decomposition of (x + 2) / [ (x + 1) (x - 2) ].

Solution: The denominator has non-repeating linear factors. Thus, by the partial fraction formulas, we get

(x + 2) / [ (x + 1) (x - 2) ] = A / (x + 1) + B / (x - 2) ... (1)

Multiplying both the sides by (x + 1) (x - 2),

(x + 2) = A (x - 2) + B (x + 1)

• On substituting x = -1: 1 = A (-3) ⇒ A = -1/3
• On substituting x = 2: 4 = B (3) ⇒ B = 4/3

On substituting the values of A and B in (1), we get

(x + 2) / [ (x + 1) (x - 2) ] = -1 / [3(x + 1)] + 4 / [3(x - 2)]

Thus, (x + 2) / [ (x + 1) (x - 2) ] = -1 / [3(x + 1)] + 4 / [3(x - 2)].

Example 2: What is the Partial Fraction Decomposition of (20x + 35)/(x + 4)²?

Solution: Given expression is (20x + 35)/(x + 4)²

(20x + 35)/(x + 4)² = [A/(x + 4)] + [B/(x + 4)²]

(20x + 35)/(x + 4)² = [A(x + 4) + B]/ (x + 4)²

On equating the numerators, we get 

20x + 35 = A(x + 4) + B

20x + 35 = Ax + 4A + B

20x + 35 = Ax + (4A + B)

On equating the coefficients,

A = 20

4A + B = 35

4(20) + B = 35

B = 35 – 80 = -45

Thus, (20x + 35)/(x + 4)² = [20/(x + 4)] – [45/(x + 4)²].

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Things to Remember

• Partial Fractions are the simple fractions formed after the decomposition of a complex algebraic fraction.
• They are used to split complex fractions into various simpler subfractions. 
• The concept of Partial Fractions is the reverse of the addition of rational expressions.
• Partial Fraction Decomposition is the process of decomposing a complex rational function as a sum of simpler rational functions.
• The degree of the numerator should be less than the degree of the denominator for Partial Fraction Decomposition.
• If the degree of the numerator is more than the degree of the denominator, it is an Improper Rational Function.
• Improper Rational Functions are first solved using long division and are then decomposed. 

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