(Keep in mind, I made this off the top of my head, so if something cancels easy, ignore it)(adsbygoogle = window.adsbygoogle || []).push({});

Let's say I had this expression:

[tex]f(x,y)=\frac{y^2-xy+1}{(x+y)(x-y)}[/tex]

I want to decompose this to:

[tex]\frac{A}{x+y} + \frac{B}{x-y}[/tex]

So i begin the process:

[tex]y^2-xy+1=A(x-y) + B(x+y)[/tex]

[tex]y^2-xy+1=x(A+B) + y(B-A)[/tex]

At this point, I can't just plug in random numbers, I need to plug in points that correspond to this point:

[tex](\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})[/tex]

So I would pick a point, say (1,1), plug it into [tex]\frac{\partial f}{\partial x}[/tex] to get the x coordinate of a point, [tex]P_1[/tex], then I would plug in the same point, (1,1), into [tex]\frac{\partial f}{\partial y}[/tex] to get the y coordinate of the [tex]P_1[/tex]. After I get two points,[tex]P_1[/tex] and [tex]P_2[/tex], I can plug them into the equation and solve the system to find A and B.

My question is, is this the right process to decompose the fraction, or am I way off?

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# Multivariable factor decomposition

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