To find the coefficient of x^4 in the expansi | vedclass

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(C) Given expression: $f(x) = \frac{3x}{(x-2)(x-1)}$.Using partial fractions: $\frac{3x}{(x-2)(x-1)} = \frac{A}{x-2} + \frac{B}{x-1}$.Solving for $A$

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$-1 < x < 1$

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(C) Given expression: $f(x) = \frac{3x}{(x-2)(x-1)}$.Using partial fractions: $\frac{3x}{(x-2)(x-1)} = \frac{A}{x-2} + \frac{B}{x-1}$.Solving for $A$ and $B$: $3x = A(x-1) + B(x-2)$.For $x=1$,$3 = B(-1) \Rightarrow B = -3$.For $x=2$,$6 = A(1) \Rightarrow A = 6$.So,$f(x) = \frac{6}{x-2} - \frac{3}{x-1} = -\frac{6}{2(1-x/2)} + \frac{3}{1-x} = -3(1-x/2)^{-1} + 3(1-x)^{-1}$.The expansion of $(1-u)^{-1}$ is valid for $|u| For $-3(1-x/2)^{-1}$,we need $|x/2| For $3(1-x)^{-1}$,we need $|x| The expansion is valid in the intersection of these intervals: $|x| < 2$ and $|x| < 1$,which is $|x| < 1$ or $-1 < x < 1$.

To find the coefficient of $x^4$ in the expansion of $\frac{3x}{(x-2)(x-1)}$,the interval in which the expansion is valid,is

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