# Algebraic Divison

Is there a way to break the left hand side expression such that it takes the the right hand side form?

$(a+b)/(c+d)=a/c+b/d+k$

Where $k$ is some expression.

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Yes, and that expression would be $(a+b)/(c+d) - a/c - b/d$. Are you looking for something less stupid or more specific? – Patrick Da Silva Jul 3 '13 at 3:17

Solve for $k$, as Patrick indicated: \begin{align} k&=\frac{a+b}{c+d}-\frac{a}{c}-\frac{b}{d}\\ &=\frac{cd(a+b)-ad(c+d)-bc(c+d)}{cd(c+d)}\\ &=\frac{acd+bcd-acd-ad^2-bc^2-bcd}{cd(c+d)}\\ &=\frac{-ad^2-bc^2}{cd(c+d)} \end{align} In the words of lots of movie cops over the years, "Move along, folks, there's nothing to see here."
@jessica: one additional thing to note is that you need $c$ and $d$ non-zero. – James Jul 3 '13 at 13:21
@James: and also $c\ne-d$, else the original expression is undefined. – Rick Decker Jul 3 '13 at 13:42
Here's an old chestnut related to your problem. Take $64/16$, cancel the 6s, and you get $4/1$ which happens to be the right answer. Too bad it doesn't work in general. – Rick Decker Jul 4 '13 at 14:05