Fibonacci Haiku: Rudolph is my Spirit Animal

Rudolph

Reindeer

Love me

That red nose

You're always my number one 

I think I make a good tenth reindeer

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SP #5: Unit J Concept 6 - Partial Fraction Decomposition with repeated factors

Be careful in this problem because you are dealing with FOUR variables rather than 3. There are some 2 variable equations in this problem so combine them together to make a 3 variable equation. Then, combine 4 variable equations to create a 3 variable equation (cancel out a variable). Combine those two 3 variable equations, and from there you can solve this system! Remember that when you are writing out the denominators for the partial fractions in the beginning, remember to count up because you are dealing with repeated factors.

SP #4 - Partial Fraction Decomposition with distinct factors

PART 1: This problem is about COMPOSING when given individual fractions to create a larger, combined fraction. We are using Algebra I/II skills here by adding fractions and making sure that our fractions have common denominators.

               Save yourself some work by multiplying the factors you are multiplying to a specific fraction. As you can see, for the first fraction, I multiplied x(x+3) to create x^2+3x, so it is a lot easier for me when I distribute that 6. Don't worry too much about the denominator, we only care about the numerator at this point.

 





PART 2: This problem is about DECOMPOSING. We are given a fraction (11x^2+40x+24/x(x+2)(x+3)) that we need to find the factors that created it. We know the factors that create the denominator and place them in individual fractions and through decomposition, we find the numerators of each fraction.

               Make sure to multiply out the factors that you are multiplying each fraction by, as stated before. It saves work in the long run. When combining like terms, remember you cross out anything "x" related because only care about the coefficients. When you create your system, make sure to see if you can take a GCF out of any of them (I divided 6 from my last equation) because we always like smaller numbers in the long run.

 

PART 3: This is simply what you type into your calculator. This is a matrix, as we have a 3 variable equation on our hands.

PART 4: This is your reduced matrix, which tells us our answers. Because the first terms represent a (IN THIS CASE, USUALLY IT IS x)and there is a 1--then we have 6 in the "answers" column/term, a=6. Our 2nd term in the 2nd row equals 1, so b=1. We already knew this plugging it in and from our system but the last row tells us that c=4.

SV#5 - Unit J Concepts 3-4: Matrices

Be careful when first approaching the system. Look for any equations that might need to be simplified. Also, when looking for your 0 for Row 3 Term 2, remember that most likely, you will need to use Row 2, because if you use Row 1, your work (your 0's in row 3) will be erased. Also, take note that because you ended up with an ordered pair as an answer, your answer is a CONSISTENT, INDEPENDENT system. If you at any point have all 0's in a row and another number that is NOT 0 in your answer column, then you have an INCONSISTENT system. If you have all 0's as every term in a row, you have a CONSISTENT, DEPENDENT system.