8.5 Partial Fraction Decomposition

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

x^2+11 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)
11=Ax^2+11A+Bx^2+Cx
11=Ax^2+Bx^2+Cx+11A
11=(A+B)x^2+Cx+11A
Equate coefficients to create a system of equations to solve
11=(A+B)x^2+Cx+11A
0x^2+0x+11=(A+B)x^2+Cx+11A
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
matrix{3}{1}{{11=11A} {0=C} {0=A+B}}

Now solve the system of equations.

The first equation is  11=11A.  Since it only has one variable, I can solve for A by hand.

11=11A
11/11={11A}/11
1=A

The second equation is 0=C .  It is already solved.

The third equation is 0=A+B.  Since I know A=1, I can substitute and solve for B.

0=A+B
0=1+B
0-1=1-1+B
-1=0+B
-1=B

Using the original set up of 11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}, substitute the values of A, B, and C.

11/{x(x^2+11)}=A/x+{Bx+C}/{x^2+11}
11/{x(x^2+11)}=1/x+{-1x+0}/{x^2+11}
11/{x(x^2+11)}=1/x+{-x}/{x^2+11}

The partial fraction decomposition of 11/{x(x^2+11)} is 1/x+{-x}/{x^2+11}