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When given a fraction with both numerator and denominator as polynomials, and where the denominator is expressed as a multiplication of factors. Can we always use the partial fractions method for splitting the overall fraction into several fractions. Or do restrictions exist for using this method, e.g The degree of the numerator must be less than the degree of the denominator, or the factors in the denominator must have real coefficients?

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If the degree of the numerator isn't less than the degree of the denominator, you can always perform (long) division first to obtain a quotient and a remainder with degree strictly less than the denominator's. You perform the partial fraction decomposition on this last fraction.

If you consider polynomials with real coefficients, completely factoring the denominator will let you end up with either linear or irreducible quadratic (i.e. with strictly negative discriminant) factors. The method always works, there are no restrictions.

You can find the more general statement (for polynomials over an arbitrary field $K$) here.

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