Express in partial fractions...?

Expect result of the form below and the equate numerators.
A/(x – 2) + (Bx + C)/(x^2 + 2) = 3x/[(x – 2)(x^2 + 2)]
A(x^2 + 2) + (Bx + C)(x - 2) = 3x equate coefficients of each power of x
x^2: A + B = 0 ………………………(1)
x: C – 2B = 3 ……………………………(2)
Constants: 2A – 2C = 0 ……………..(3)
Substitute C = A from (3) and B = - A from (1) into (2)
A – (-2A) = 3
A = 1 = C and B = -1

Result can be written as 1/(x – 2) - (x - 1)/(x^2 + 2),
or as 1/(x – 2) - x /(x^2 + 2) + 1/(x^2 + 2)

A(2 + x²) + (Bx + C)(x - 2) = 3x


This method may be new to you, but expand:


2A + Ax² + Bx² - 2Bx + Cx - 2C = 3x


Now prepare to factor by grouping. In other words group all the x-terms near each other, all the coefficients together and all the x² 's like so:


Ax² + Bx² - 2Bx + Cx + 2A - 2C


Factor out the x 's for the groups that contain x ' s.


(A + B)x² + (- 2B + C)x + 2A - 2C = 3x


Okay, now you may be wondering why we did this. Well take a very close look at the left side of the equation and compare it to the right side.

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How many x² 's take place on the right side of the equation? None, so (A + B) *must* equal zero.


There is an x-term on the right side of the equation and it dictates that the coefficient is going to be three, therefore - 2B + C *must* be equal to three in order to match the terms on the right side of the equation.


There are NO constants on the right side of the equation, therefore + 2A - 2C must be equal to zero.

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⇒ Through these three observations we can make a system of equations! Written below:


2A - 2C = 0


A + B = 0


- 2B + C = 3


Can you take it from here? if you solve this system of equations correctly you'll end up with:


A = 1


B = - 1


C = 1


Plug these in accordingly and you'll find that your decomposed fraction is:


1/(x - 2) + (- x + 1) / (x² + 2)


This is your final answer. If you add them together you'll see that it'll end up being the original fraction you started out with. This ensures the answer is correct. Here's a link of the addition so you can be certain: https://www.symbolab.com/solver/step-by-step/1%2F%5Cleft(x%20-%202%5Cright)%20%2B%20%5Cleft(-%20x%20%2B%201%5Cright)%20%2F%20%5Cleft(x%C2%B2%20%2B%202%5Cright)