SP#5: Unit J Concept 6: Partial Fraction Decomposition (Repeated Factors ver.)

*FYI: The step numbers do not necessarily correlate with the numbers in the image.*

Step 1: Seperate your denominator factors into fractions- in this case, one of the factors is cubed, so when writing the fractions out remember to count up the factors. [1]
Step 2: Find a common denominator. In this case, you simply have to multiply the numerator by the factors that aren't already in the denominator. Then once you've multiplied and FOIL'd all the numbers and letters, set it equal to your original equation.
Step 3: Seperate the like terms into 4 different equations. At this point, you can take out the 'x cubed', 'x squared', and 'x' values and leave just the coefficients. [2]
Step 4: Two of your equations should have 2 terms and the other two should have 4 terms. Take the latter two equations and use elimination to get rid of a term. In this case, I multiplied the purple equation by two, then added it to the other 4 term equation to get rid of the D term. [3a]
Step 5: Now you have a 3 term equation, but no other one lying around ready to use. Add the 2 term equations together to make another 3 term equation. [3b] Use elimination again to get rid of another value- hopefully you should see where I'm going with this. In this case, I multiplied the 3 term equation by 4 to get rid of the C term.
Step 6: Now you have a 2 term equation. Take any one of your first 2 term equations and use elimination again to get rid of a term. In this case, I multiplied the pink equation by 4 to get rid of the B term, leaving me with the A term.
Step 7: Solve for that final term, then back substitute until you have solved for all the terms. That's it, you're done~!