Values tried for partial fraction decomposition

Question

I'll explain my question with the following example from wikipedia. Suppose, we have a function: $$ f(x)=\frac{1}{x^2+2x-3} $$ Here, the denominator splits into two distinct linear factors: $$ q(x)=x^2+2x-3 = (x+3)(x-1) $$ so we have the partial fraction decomposition $$ f(x)=\frac{1}{x^2+2x-3} =\frac{A}{x+3}+\frac{B}{x-1} $$ Multiplying through by $x^2 + 2x − 3$, we have the polynomial identity $$ 1=A(x-1)+B(x+3) $$ Substituting $x = −3$ into this equation gives $A = −1/4$, and substituting $x = 1$ gives $B = 1/4$, so that $$ f(x) =\frac{1}{x^2+2x-3} =\frac{1}{4}\left(\frac{-1}{x+3}+\frac{1}{x-1}\right) $$ My doubt: For $x = -3$ and $x = 1$, our function $f(x)$ is undefined. So, is it valid to substitute these values for $x$? If it is, then why? And if it isn't, then why do these values work?

Answer 1

There really isn't a problem - this is a convenient trick which works. You can solve the identity as well by equating coefficients, and solving linear equations. It can be treated as a polynomial identity and solved in an entirely symbolic way. If there is a solution, it must be this one. It is when you regard the symbolic expression as a function that you get issues about undefined values.

Here is another way of looking at eliminating $A$, which may help. Set $x=1+t$ and $x=1-t$ to obtain:

$$1=At+B(4+t)$$ $$1=-At+B(4-t)$$

Now add these two equations and divide by $2$ to obtain $$1=4B$$

This is exactly what you would have got by putting $x=1$ in the original equation.