## Saturday, November 16, 2013

### SP#5: Unit J Concept 6 - Partial Fraction Decomposition With Repeated Factors

Make sure to pay special attention to when you are simplifying and foiling. A single mistake could make your entire answer wrong. Also, check to see that you are solving the system of 4 equations correctly, that you grouped the right like terms and eliminated/substituted properly. There will be a lot of elimination involved with this problem, since there will be 4 equations and 4 variables.

Problem: (x^2 - 2)/(x-2)(x+1)^3

1. Since denominator is already factored fully, find the common denominator.
2. Separate each factor into a separate fraction with and assigned letter as each of its numerator. Since one of the factors repeat, you must count up the powers and include the factor as many times as the exponent.
3. Multiply each part(numerator and denominator) by what is missing from the common denominator.
4. Simplify. Set numerator equal to the numerator of the problem because the common denominator doesn't matter now.
4. Group all like terms together and set it equal to the corresponding like term on the other side.
5. Remove the coefficients, but leave the coefficients as they are in the equation. You end up with a system of 4 different equations.

6. Solve resulting system of equations by using elimination. We want to get rid of D first since there are only two equations with D, so that we end up with a new 3 variable equation.
7. Combine the first two equations to make another 3 variable equation.
8. Use elimination on the 2 new equations to make a new 2 variable equation.
9. Use elimination on the new 2 variable equation with either one of the 2 original 2 variable equation to calculate the value of one variable.
10. Substitute value back into the other equations to calculate the remaining values of the other variables.

11. Substitute values of variables back into original split up fractions with the letters.