SOLUTION: 2y+1 ___________ y^2-y-6

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Question 196914This question is from textbook
: 2y+1
___________
y^2-y-6 This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that you want to perform a partial fraction decomposition...


%282y%2B1%29%2F%28y%5E2-y-6%29 Start with the given expression.


%282y%2B1%29%2F%28%28y%2B2%29%28y-3%29%29 Factor the denominator



Since both factors are linear and are of multiplicity 1, this means that the fraction decomposes into:

%282y%2B1%29%2F%28%28y%2B2%29%28y-3%29%29=A%2F%28y%2B2%29%2BB%2F%28y-3%29


Multiply EVERY term by the LCD %28y%2B2%29%28y-3%29 to clear out the fractions.


2y%2B1=A%28y-3%29%2BB%28y%2B2%29 Simplify


2y%2B1=Ay-3A%2BBy%2B2B Distribute


2y%2B1=Ay%2BBy-3A%2B2B Rearrange the terms


2y%2B1=%28A%2BB%29y-3A%2B2B Combine like terms.



Since the "y" coefficient on the right side is A%2BB, this means that A%2BB=2. Also, the constant term -3A%2B2B=1



So this means that we have the system:

system%28A%2BB=2%2C-3A%2B2B=1%29


Solve this system of equations (using any method) to get the solutions:

A=3%2F5 and B=7%2F5


Now plug these values back into the original equation to get:






And simplify


%282y%2B1%29%2F%28%28y%2B2%29%28y-3%29%29=3%2F%285%28y%2B2%29%29%2B7%2F%285%28y-3%29%29

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Answer:

So %282y%2B1%29%2F%28y%5E2-y-6%29=3%2F%285%28y%2B2%29%29%2B7%2F%285%28y-3%29%29