Example: Find out the partial fraction decomposition of following.

             8x - 42 / x²+ 3x -18

Solution

The primary thing to do is factor out the denominator as much as we can.

                          8x - 42 / x²+ 3x -18 = 8x - 42 / ( x+ 6) ( x - 3)

Thus, by comparing to the table above it seems like the partial fraction decomposition has to look like,

                                      8x - 42 / x²+ 3x -18 = A/(x+6) + B/(x-3)

Note that we've got distinct coefficients for each term as there is no cause to think that they will be the similar.

Now, we have to determine the values of A & B. The first step is to in fact add the two terms back up. Usually this is simpler than it might seem to be.  Recall that first we required the least common denominator; however we've already got that from the original rational expression.  In this case it is,

                                           LCD = ( x + 6) ( x - 3)

Now, only look at each of term and compare the denominator to the LCD.  Multiply the numerator & denominator through whatever is missing then add.  In this case this gives,

8x - 42/ x²+ 3x -18

=            ( A (x - 3)/( x+ 6) ( x - 3) + (B ( x + 6) /( x+ 6) ( x - 3) +(A ( x - 3) + B ( x + 6) /( x+ 6) ( x - 3)

We require values of A & B so that the numerator of the expression on the left is the simialr as the numerator of the term on the right.  Or,

8x - 42 = A ( x - 3) + B ( x + 6)

It has to be true regardless of the x that we plug into this equation.  As illustrious above there are various ways to do this.  One method will always work, however can be messy and will frequently require knowledge which we don't have yet.  The other way will not always work, however while it does it will greatly reduce the amount of work needed.

In this set of instance the second (and easier) method will always work thus we'll be using that here. Here we are going to make utilization of the fact that this equation have to be true regardless of the x that we plug in.

Thus let's pick an x, plug it in & see what happens. For no clear reason let's try plugging in x= 3 .   Doing this provides,

8 (3) - 42 = A (3 - 3) + B (3 + 6)

-18 = 9B

-2 =B

Can you see why we select this number?  By choosing x= 3 we got the term including A to drop out & we were left with a simple equation which we can solve for B.

Now, we could also select x= -6 for exactly the same cause.  Here is what happens if we utilize this value of x.

8 ( -6) - 42 = A (-6 - 3) + B ( -6 + 6)

-90 = -9 A

10 = A

Thus, by correctly picking x we were capable to quickly & easily get the values of A & B.  Thus, all that we have to do at this point is plug them in to finish the problem.  Following is the partial fraction decomposition for this part.

8x - 42/ x²+ 3x -18

=  (10 /(x+ 6) + (-2 / (x - 3) = (10 /(x+ 6) - (2 / (x - 3)

Notice, we moved the minus sign on the second term down to make the addition a subtraction. We will always do that.