Question 997722
i did a little research and i think i understand the process.


start with 8 / (y^3 - 4y)


factor the denominator to get:


8 / (y * (y+2) * (y-2))


i believe they call it that you decompose the fraction to get:


8 / (y * (y+2) * (y-2)) = A/y + B/(y+2) + C/(y-2)


you now multiply both sides of this eqaution by (y * (y+2) * (y-2) to get rid of the fractions.


you are left with:


8 = A*(y+2)*(y-2) + B*y*(y-2) + C*y*(y+2)


you now simplify the right side of the equation to get:


8 = A*(y^2-4) + B*(y^2-2y) + C*(y^2+2y)


you now simplify the right side further by performing the operations indicated to get:


8 = A*y^2 - 4a + B*y^2 - 2By + C*y^2 + 2Cy


you now combine like terms and reorder the terms in descending order to get:


8 = A*y^2 + B*y^2 + C*y^2 - 2By + 2Cy - 4A


you now factor out the common terms to get:


8 = (A + B + C)*y^2 + (-2B + 2C)*y - 4A.


you now take your expression on the left and add in the missing terms to get:


8 = 0*y^2 + 0*y + 8


your equation becomes:


0*y^2 + 0*y + 8 = (A + B + C)*y^2 + (-2B + 2C)*y - 4A.



you now equate the coefficients from the right side of the equation and the left side of the equation for the terms with the same variables.


A+B+C = 0 for the y^2 term.
-2B + 2C = 0 for the y term.
-4A = 8 for the constant term.


-4A = 8 leads to A = -2


-2B + 2C = 0 leads to B = C


A+B+C = 0 becomes -2 + B + C = 0


since B = C, this becomes either -2 + 2B = 0, or -2 + 2C = 0


either way, you get B = 1 and C = 1


you are left with:
A = -2
B = 1
C = 1


your solution is:


8 / (y^3 - 4y) = -2/y + 1/(y+2) + 1/(y-2)


you can confirm the solution is correct by picking any number to be equal to y and then seeing if the equation is true.


i picked y = 17 


i got:


8 / (y^3 - 4y) = -2/y + 1/(y+2) + 1/(y-2) becomes:


8 / (17^3 - 4*17) = -2/17 + 1/19 + 1/15


this was resolved to get:


.0016511868 = .0016511868


since they are the same, this confirmed the solution is correct.


here's a reference from the web you might find useful.


<a href = "https://www.youtube.com/watch?v=7cgOf3alK40" target = "_blank">https://www.youtube.com/watch?v=7cgOf3alK40</a>


patrickJMT has numerous examples you can look at if you want to dive into it further, including examples where the denominator doesn't factor.


there are also numerous other youtube videos that talk about the same subject.


this is about as far as i want to go into it so good luck with the rest.