In the proper fraction ,the denominator in factor of the form
and
the partial fraction decomposition must include following sum of m and n fractions respectively,
and
.
The rational expression is .
In the above expression is proper fraction and its denominator is .
The factor form of denominator, ,so m = 2.The form of decomposition of expression is
.
(Write form of decomposition)
(Multiply each side by
)
(Multiply)
(Combine like terms)
The two polynomials are equal if and only if the coefficient like terms are equal, you can equate the coefficient of like terms o.So, we can write the system of linear equations are
\ (Equation 1)
(Equation 2)
(Equation 3)
From the equation 3 the value of is substitute in equation 2 then produces a new second equations.So the new system of equation are
(Equation 1)
(Equation 2)
(Equation 3)
Add first equation to the second equation to obtain .
The value of substitute in first equation to obtain
.
From third equation to obtain .
So, the partial fraction decomposition is .
The partial fraction decomposition is .