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.
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It's easy to see that a(bx + d) - b(ax + c) = ad - bc. Dividing both sides by [(ax + c)(bx + d)] then gives a/(ax + c) - b/(bx + d) = (ad - bc)/[(ax + c)(bx + d)].
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Also note that (ad - bc)/[(a(-d/b) + c)(bx + d)] + (ad - bc)/[(ax + c)(b(-c/a) + d)] = a/(ax + c) - b/(bx + d) = (ad - bc)/[(ax + c)(bx + d)]. This elegant pattern persists throughout the theory of partial fractions, rendering it memorable, and its application simple.
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Replying to @MathPrinceps
Are these secretly facts about SL(2), the group of fractional linear transformations? (I'm just groping around here.)
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Yes, indeed they are. At the bottom of all this is the Euclidean algorithm, which is our natural means of navigation through what John Horton Conway calls the "topograph" of the rational numbers. The theory of partial fractions is inseparable from this object and its structure.
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