Splitting a fraction

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Splitting a fraction apart, the partial-fraction decomposition, breaks a single rational function into a sum of simpler ones, one for each factor of the denominator. It is the reverse of adding fractions over a common denominator, and it is what turns a rational function into a form that can be integrated. Like the other algebra forms, this does not need a trailing arrow.

apart is the other direction: it breaks a rational function into the simple fractions that add up to it, one for each factor of the denominator.

apart((3x+5)/(x^2-1)) // 4/(x-1)-1/(x+1)
apart((x^2+1)/(x^3-x)) // -1/x+1/(x-1)+1/(x+1)

A repeated factor gets one piece per power of it, and a fraction that is not proper keeps its polynomial part out front.

apart(1/(x*(x+1)^2)) // 1/x-1/(x+1)-1/(x+1)^2
apart((x^3+1)/(x^2-1)) // x+1/(x-1)

A denominator with nothing to split, because it is already irreducible, comes back as it was.

apart((2x+3)/(x^2+x+1)) // (2x+3)/(x^2+x+1)

This is what makes a rational function integrable. See calculus.