Find constants A and B such that

\(\frac{x - 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1} \)

for all x such that \(x \neq 1\) and \(x \neq 2\).

Letting x go to 2, we find that A + 2B = 4. Letting x go to -1, we find 2A - B = 3. Therefore, A = 2 and B = -1.

x does not equal 1 or 2 though. I am confused. Can you explain?

If you let x go infinity, then you get 0 = A + B.

If you let x = 0, then you get 7/2 = A/(-2) + B.

The solution to this system is then A = -7/3, B = 7/3, so (A,B) = (-7/3,7/3).