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How do you express (2x-1)/((x-1)^3(x-2))...

How do you express `(2x-1)/((x-1)^3(x-2))` in partial fractions?

Text Solution

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`frac{2 x-1}{(x-1)^{3}(x-2)}=frac{A}{(x-1)^{3}}+frac{B}{(x-1)^{2}}+frac{C}{x-1}+frac{D}{x-2}`

`=frac{A(x-2)+B(x-1)(x-2)+C(x-1)^{2}(x-2)+D(x-1)^{3}}{(x-1)^{3}(x-2)}`

The denominators are the same, we compare the numerators

`2 x-1=A(x-2)+B(x-1)(x-2)+C(x-1)^{2}(x-2))+D(x-1)^{3}`

Let `x=1, Rightarrow, 1=-A, Rightarrow, A=-1`

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