Partial Fraction Decomposition

Question

(-x²+3x+8)/(x³+4x²+4x)

Answer 1

When having partial fractions, you want to convert the given single fraction into a sum of smaller fractions.

 

First, we factor out the denominator.

 

x(x² + 4x + 4) = x(x + 2)(x + 2)

 

 

Now, we write the given fraction with the factored denominator with an equivalent sum of fractions.

 

-x² + 3x + 8                  A                   B                   C

_____________   =   _______  +  ________  +   _______

x(x + 2)(x + 2)             x                x + 2              x + 2

 

 

 

                                 A(x + 2)(x + 2) + Bx(x + 2) + Cx(x + 2)

                         =    _____________________________________

                                         x(x + 2)(x + 2)

 

 

 

 

                         =   Ax² + 4Ax + 4A + Bx² + 2Bx + Cx² + 2Cx

                              ___________________________________

                                               x(x + 2)(x + 2)

 

 

 

                         =   x²(A + B + C) + x(4A + 2B + 2C) + 4A

                             __________________________________

                                                 x(x + 2)(x + 2)

 

 

Now, we equate coefficients.

 

x² :           A + B + C = -1

x  :         4A + 2B + 2C = 3

const:     4A = 8

 

 

You now have a system of equations.  Solve for A, B, and C using the equations.

Answer 2

The denominator, in factored form, is x(x+2)²

 

So, the partial fraction decomposition is A/x + B/(x+2) + C/(x+2)²

 

Simplifying, we get [A(x+2)² + Bx(x+2) + Cx]/[x(x+2)²]

 

Expanding the numerator and combining like terms, we have:

 

   (A+B)x² + (4A+2B+C)x + 4A 

 

So, (A+B)x² + (4A+2B+C)x + 4A = -x² + 3x + 8

 

Equating coefficients of like powers, we have       A+B = -1

                                                              4A+2B+C = 3

                                                                        4A = 8

 

So, A = 2, B = -3, C = 1

 

Thus, the partial fraction decomposition is:

 

     2/x - 3/(x+2) + 1/(x+2)²