# 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.

• 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,
= .

## Solved Examples for Partial Fractions

Find the partial fraction decomposition of .
Choices:
A.
B.
C.
D. none of the above
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, = .

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