Partial Fraction

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. 

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:

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: \(\frac{7x + 4}{(x + 1)(x + 2)} = \frac{-3}{x + 1} + \frac{10}{x + 2} \)

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.

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:

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.

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.