(a) Using the substitution \(u = x – 2\), write \(\frac{x^{3} + 5}{(x – 2)^{4}}\) as an expression in terms of u.
(b) Using the answer in (a), express \(\frac{x^{3} + 5}{(x – 2)^{4}}\) in partial fractions.
Explanation
(a) \(u = x - 2 \implies x = u + 2\)
\(\frac{x^{3} + 5}{(x - 2)^{4}}\) becomes \(\frac{(u + 2)^{3} + 5}{u^{4}}\)
= \(\frac{u^{3} + 6u^{2} + 12u + 8 + 5}{u^{4}}\)
= \(\frac{u^{3} + 6u^{2} + 12u + 13}{u^{4}}\)
= \(\frac{1}{u}\) + \(\frac{6}{u^{2}}\) + \(\frac{12}{u^{3}}\) + \(\frac{13}{u^{4}}\)
(b) \(\frac{x^{3} + 5}{(x - 2)^{4}}\) = \(\frac{1}{x - 2}\) + \(\frac{6}{(x - 2)^{2}}\) + \(\frac{12}{(x - 2)^{3}}\) + \(\frac{13}{(x - 2)^{4}}\)