The system of equations to solve for the unknowns is [tex]A= 2[/tex], [tex]-6A + B = 0[/tex] and [tex]9A- 3B + C = 1[/tex]
The given parameters are:
[tex]\frac{2x^2 + 1}{(x - 3)^3} = \frac{A}{x - 3} + \frac{B}{(x - 3)^2} + \frac{C}{(x - 3)^3}[/tex]
[tex]2x^2 + 1 =A(x -3)^2 + B(x -3) +C[/tex]
Start by opening the brackets
[tex]2x^2 + 1 =A(x^2 -6x + 9) + Bx -3B +C[/tex]
This gives
[tex]2x^2 + 1 =Ax^2 -6Ax + 9A + Bx -3B +C[/tex]
Next, collect the like terms
[tex]2x^2 + 1 =Ax^2 -6Ax + Bx + 9A -3B +C[/tex]
Next, compare both sides of the equation.
This gives
[tex]Ax^2 = 2x^2[/tex]
[tex]-6Ax + Bx = 0[/tex]
[tex]9A- 3B + C = 1[/tex]
Lastly, cancel out the variable x
[tex]A= 2[/tex]
[tex]-6A + B = 0[/tex]
[tex]9A- 3B + C = 1[/tex]
Hence, the system of equations to solve for the unknowns is [tex]A= 2[/tex], [tex]-6A + B = 0[/tex] and [tex]9A- 3B + C = 1[/tex]
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