Resolve \(\frac{3x - 1}{(x - 2)^{2}}, x \neq 2\) into partial fractions.
A. \(\frac{x}{2(x - 2)} - \frac{5}{(x - 2)^{2}}\)
B. \(\frac{5}{(x - 2)} + \frac{x}{2(x - 2)^{2}}\)
C. \(\frac{1}{2(x - 2)} + \frac{5x}{2(x- 2)^{2}}\)
D. \(\frac{-1}{2(x - 2)} + \frac{8x}{2(x - 2)^{2}}\)
Correct Answer: C
Explanation
\(\frac{3x - 1}{(x - 2)^{2}} = \frac{A}{(x - 2)} + \frac{Bx}{(x - 2)^{2}}\)
\(\frac{3x - 1}{(x - 2)^{2}} = \frac{A(x - 2) + Bx}{(x - 2)^{2}}\)
Comparing, we have
\(3x - 1 = Ax - 2A + Bx \implies -2A = -1; A + B = 3\)
\(\therefore A = \frac{1}{2}; B = \frac{5}{2}\)
= \(\frac{1}{2(x - 2)} + \frac{5x}{2(x - 2)^{2}}\)
