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Why is it that when you have a repeated root in the denominator of a rational proper function, you include different powers of the same root in the function's partial fraction expansion?

For example,

[tex] \frac{x^2 + 4x + 7}{(x-3)^3} = \frac{k_1}{(x-3)} + \frac{k_2}{(x-3)^2} + \frac{k_3}{(x-3)^3}[/tex]

why do you do this?

For example,

[tex] \frac{x^2 + 4x + 7}{(x-3)^3} = \frac{k_1}{(x-3)} + \frac{k_2}{(x-3)^2} + \frac{k_3}{(x-3)^3}[/tex]

why do you do this?

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