SP#4: Unit J Concept 5 - Partial Fraction Decomposition with Distinct Linear Factors

Make sure to pay special attention when you are decomposing or composing the fraction. Check to see if you foiled and distributed correctly, and if you grouped all like terms. In addition, when solving, make sure your equations are correct, your matrix is set up correctly, and the values are correctly substituted back into the split up fractions.

Problem: -5/x + 2/(x+3) + 1/(x-2)

Part 1: Composing Fractions

1. Find the common denominator. Multiply each part by what it is missing in the numerator and the denominator.

2. Foil out the factors first and then distribute the numerator. Keep the denominator factored.

3. Simplify the expression by combining all like terms. 

Part 2: Decomposing 

Problem from Part 1: (-2x^2 - 6x + 30)/(x(x+3)(x-2))

1. Since denominator is already factored fully, separate each factor into a fraction assigned with a different letter. (Doesn't matter what letter as long as it's not x since it's already being used)

2. Find the common denominator. Multiply each fraction (numerator and denominator) by what it is missing from the common denominator.

3. Simplify by foiling out the factors first and then distributing the numerator. Keep the denominator factored. 

4. Set the numerator equal to the numerator of the problem. (Denominator doesn't matter since common now. Can ignore it.)

5. Group like terms with letters together and set it equal to the like terms on the other side. 

6. Take out the x's so that only the coefficients remain. (Letters stay too.)

Part 3: Solving System of Equation with Matrices

1. Take the coefficients and set them up in a matrix.

2. Plug into graphing calculator. (2nd Matrix, Edit, Select any letter, Input coefficients of original equations, Quit, 2nd Matrix, Press rref( feature, 2nd Matrix, Select edited matrix, Close Parenthesis, Enter)

3. Calculator will give Reduced Row-Echelon Form. Record values of letters.

4. Plug values of letters back into split up fractions from the beginning. (Should end up with original problem.)