8.5 Partial Fraction Decomposition

Example: Find the partial fraction decomposition for the rational expression.

$11/{x(x^2+11)}$

Solution:

The rational expression	$11/{x(x^2+11)}$
Factor the denominator completely	$11/{x(x^2+11)}$
Write the expression with each factor as a separate fraction.	$11/{x(x^2+11)}=?/x+?/{x^2+11}$
Decide what type of expression to put in the numerator x is linear so we put a constant in the numerator is a quadratic so we put a linear expression in the numerator	$11/{x(x^2+11)}=?/x+?/{x^2+11}$ $11/{x(x^2+11)}=A/x+?/{x^2+11}$ $11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}$
Now that the set up is done we need to solve for the unknowns A, B and C	$11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}$
Multiply both sides of the equation by the common denominator ${x(x^2+11)}$ and simplify by canceling out common factors	$11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}$ ${x(x^2+11)}11/{x(x^2+11)}={x(x^2+11)}(A/x+{Bx+C}/{x^2+11})$ $11={x(x^2+11)}{A/x}+{x(x^2+11)}{Bx+C}/{x^2+11}$ $11={(x^2+11)}{A}+x(Bx+C)$
Use the distributive property and collect the like terms with the x’s	$11={(x^2+11)}{A}+x(Bx+C)$
Equate coefficients to create a system of equations to solve
On the left 11 is the constant term. There is no linear term so the coefficient is zero and there is no quadratic term so the coefficient is zero. On the right the constant (no x’s) is 11A, the linear coefficient is C and the quadratic coefficient is A+B.	System of equations from equating the coefficients