a/(ax + c) - b/(bx + d) = (ad - bc)/[(ax + c)(bx + d)]. This one identity suffices to underpin the entire theory of partial fractions. Note that a(-d/b) + c = (-1/b)(ad - bc) b(-c/a) + d = (+1/a)(ad - bc) These truths are neither accidental nor insignificant.
Also, perhaps I ought to have noted that 1/[(ax + c)(bx + d)] = 1/[(a(-d/b) + c)(bx + d)] + 1/[(ax + c)(b(-c/a) + d)]. From the above it then follows that (mx + n)/[(ax + c)(bx + d)] = (m(-d/b) + n)/[(a(-d/b) + c)(bx + d)] + (m(-c/a) + n)/[(ax + c)(b(-c/a) + d)]
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So here's the general story: P(x)/Q(x) = [(mx + n) ... (px + q)]/[(ax + c) ... (bx + d)] = [(m(-d/b) + n) ... (p(-d/b) + q)]/[(a(-d/b) + c) ... (bx + d)] + ... [(m(-c/a) + n) ... (p(-c/a) + q)]/[(ax + c) ... (b(-c/a) + d)], provided deg(P) < deg(Q).
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