The system of equations to solve for the unknowns is
,
and ![9A- 3B + C = 1](https://tex.z-dn.net/?f=9A-%203B%20%2B%20C%20%3D%201)
<h3>How to determine the system of equations?</h3>
The given parameters are:
![\frac{2x^2 + 1}{(x - 3)^3} = \frac{A}{x - 3} + \frac{B}{(x - 3)^2} + \frac{C}{(x - 3)^3}](https://tex.z-dn.net/?f=%5Cfrac%7B2x%5E2%20%2B%201%7D%7B%28x%20-%203%29%5E3%7D%20%3D%20%5Cfrac%7BA%7D%7Bx%20-%203%7D%20%2B%20%5Cfrac%7BB%7D%7B%28x%20-%203%29%5E2%7D%20%2B%20%5Cfrac%7BC%7D%7B%28x%20-%203%29%5E3%7D)
![2x^2 + 1 =A(x -3)^2 + B(x -3) +C](https://tex.z-dn.net/?f=2x%5E2%20%2B%201%20%3DA%28x%20-3%29%5E2%20%2B%20B%28x%20-3%29%20%2BC)
Start by opening the brackets
![2x^2 + 1 =A(x^2 -6x + 9) + Bx -3B +C](https://tex.z-dn.net/?f=2x%5E2%20%2B%201%20%3DA%28x%5E2%20-6x%20%2B%209%29%20%2B%20Bx%20-3B%20%2BC)
This gives
![2x^2 + 1 =Ax^2 -6Ax + 9A + Bx -3B +C](https://tex.z-dn.net/?f=2x%5E2%20%2B%201%20%3DAx%5E2%20-6Ax%20%2B%209A%20%2B%20Bx%20-3B%20%2BC)
Next, collect the like terms
![2x^2 + 1 =Ax^2 -6Ax + Bx + 9A -3B +C](https://tex.z-dn.net/?f=2x%5E2%20%2B%201%20%3DAx%5E2%20-6Ax%20%2B%20Bx%20%2B%209A%20%20-3B%20%2BC)
Next, compare both sides of the equation.
This gives
![Ax^2 = 2x^2](https://tex.z-dn.net/?f=Ax%5E2%20%3D%202x%5E2)
![-6Ax + Bx = 0](https://tex.z-dn.net/?f=-6Ax%20%2B%20Bx%20%3D%200)
![9A- 3B + C = 1](https://tex.z-dn.net/?f=9A-%203B%20%2B%20C%20%3D%201)
Lastly, cancel out the variable x
![A= 2](https://tex.z-dn.net/?f=A%3D%202)
![-6A + B = 0](https://tex.z-dn.net/?f=-6A%20%2B%20B%20%3D%200)
![9A- 3B + C = 1](https://tex.z-dn.net/?f=9A-%203B%20%2B%20C%20%3D%201)
Hence, the system of equations to solve for the unknowns is
,
and ![9A- 3B + C = 1](https://tex.z-dn.net/?f=9A-%203B%20%2B%20C%20%3D%201)
Read more about partial fraction at:
brainly.com/question/26689595
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