PARTIAL FRACTIONS

Definition Of Partial Fractions

When a proper rational expression is decomposed into a sum of two or more rational expressions, it is known as Partial Fractions.

More About Partial Fractions

It is used in integrating rational fractions in calculus and finding the inverse Laplace transform.
In partial fractions the degree of numerator is less than the degree of the denominator.

Examples of Partial Fractions

The rational function  can be decomposed into partial fractions in the following way:
First decompose the fraction into linear factors as 
 = 
On simplification, x - 4 = A(x + 4) + B(x)
Now, by comparing the coefficients of like terms on both sides, we get,
A + B = 1, 4A = - 4.
On solving the equations, we get, A = - 1, B = 2.
By substituting the values of A and B, we get,
 = .

Video Examples: Partial Fraction Expansion 1

Solved Example on Partial Fractions

Ques: Find the partial fraction decomposition of  .

Choices:

A. 
B. 
C. 
D. none of the above
Correct Answer: C

Solution:

Step 1:  = 
Step 2:  = 
Step 3: = 5 = A(x + 3) + B(x + 2) 
Step 4: Then A + B = 0; 3A + 2B = 5 [Compare the coefficients of like terms on both sides.]
Step 5:   = 
C X = D [Write system of equations in the matrix form as CX = D.]
Step 6: X = C-1D
Step 7:  =   [Use Matrix Inversion method.]
Step 8: = -1  [Inverse of =  .]
Step 9: = 
Step 10: So, A = 5 and B = - 5
Step 11: So,  = .