# How do I find the partial fraction decomposition of #(x^4)/(x^4-1)# ?

By Partial Fraction Decomposition, we can write

Let us look at some details.

By rewriting a bit,

Let us find the partial fractions of

by factoring out the denominator,

by splitting into the partial fraction form,

by taking the common denominator,

by simplifying the numerator,

Since the numerator is originally 1, by matching the coefficients,

By adding (1) and (3),

By adding (2) and (4),

By adding (5) and (6),

By plugging (7) into (5),

By plugging (7) and (8) into (1),

By plugging (7) and (8) into (2),

By (5), (6), (9), and (10),

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To find the partial fraction decomposition of (\frac{x^4}{x^4-1}), first factor the denominator as ((x^2-1)(x^2+1)). Then, express (\frac{x^4}{x^4-1}) as (\frac{A}{x-1} + \frac{B}{x+1} + \frac{Cx+D}{x^2+1}), where (A), (B), (C), and (D) are constants. Then solve for (A), (B), (C), and (D).

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