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.