Question

Can someone help me with partial fraction decomposition? The problem is -4x-9/(x-9)^2

Answer 1

Ok, well we're looking for some A and B such that:

\[\frac{-4x - 9 }{(x-9)^2} = \frac{A}{x-9} + \frac{B}{(x-9)^2}\]
\[\implies -4x -9 = A(x-9) + B\]
Taking the limit as x goes to 9 we have:
\[lim_{x\rightarrow 9} [-4x -9] = lim_{x\rightarrow 9}[A(x-9) + B]\]\[\implies B = -4(9) - 9 = -5(9) = -45\]Then plugging that value back in for B we can find A\[\implies -4x - 9 = A(x-9) - 45\]\[\implies -4x - 9 + 45 = A(x-9)\]\[\implies A = \frac{-4x + 36}{x-9} = \frac{-4(x-9)}{(x-9)} = -4\]

Therefore:
\[\frac{-4x-9}{(x-9)^2} = \frac{-4}{x-9} + \frac{-45}{(x-9)^2}\]