Fibonacci Haiku: Time is Precious

Time

Fleeting

Always moving

Infinite yet limited

The past cannot be undone

The future is uncertain, the present a gift

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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.

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.)

SV#5: Unit J Concepts 3-4 - Solving 3 Variable Systems with Matrices

To view my video, please click here.

Make sure to pay special attention to your matrices and that all the numbers you write are correct because even if you forget something like a negative sign, then your entire answer will be wrong. You should also be careful about your elementary row operations to make sure your able to get Row-Echelon form and check your graphing calculator to check if your answer is indeed correct.