Partial Fractions: 1/(x⁴ − 1)
1/(x⁴ − 1) decomposes into (1/4)/(x - 1) + (-1/4)/(x + 1) + (-1/2)/(x^2 + 1). A worked mixed factors example with the factorisation, the coefficient system, and an editable calculator.
1/(x⁴ − 1) = (1/4)/(x - 1) + (-1/4)/(x + 1) + (-1/2)/(x^2 + 1). x⁴ − 1 splits into (x − 1)(x + 1)(x² + 1): two real roots and one irreducible quadratic.
e.g. 3x+11
Expanded or factored, e.g. (x-2)(x+3) or x^2+x-6
The decomposition
1/(x⁴ − 1) = (1/4)/(x - 1) + (-1/4)/(x + 1) + (-1/2)/(x^2 + 1)
Denominator factors: (x - 1) × (x + 1) × (x^2 + 1)
This is a mixed factors problem. x⁴ − 1 splits into (x − 1)(x + 1)(x² + 1): two real roots and one irreducible quadratic.
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Frequently asked questions
How does 1/(x⁴ − 1) decompose? ▾
1/(x⁴ − 1) = (1/4)/(x - 1) + (-1/4)/(x + 1) + (-1/2)/(x^2 + 1). x⁴ − 1 splits into (x − 1)(x + 1)(x² + 1): two real roots and one irreducible quadratic.
How was the denominator factored? ▾
It factors as (x - 1) × (x + 1) × (x^2 + 1), which sets the shape of the decomposition.
How do I check the answer? ▾
Add the resulting fractions back over a common denominator — you should recover the original expression. Substituting a convenient value of x into both sides is a quicker spot check.
About this Calculator
1/(x⁴ − 1) decomposes into (1/4)/(x - 1) + (-1/4)/(x + 1) + (-1/2)/(x^2 + 1). A worked mixed factors example with the factorisation, the coefficient system, and an editable calculator.