Theory question 103
Polynomials(a) Using the substitution u = x − 2, write (x3 + 5)/(x − 2)4 as an expression in terms of u.
(b) Using the answer in 3(a), express (x3 + 5)/(x − 2)4 in partial fractions.
Show the model answer
Model answer
(a) x = u + 2: ((u + 2)³ + 5)/u⁴ = (u³ + 6u² + 12u + 13)/u⁴ = 1/u + 6/u² + 12/u³ + 13/u⁴.
(b) 1/(x − 2) + 6/(x − 2)² + 12/(x − 2)³ + 13/(x − 2)⁴.
Explanation
Expand (u + 2)³ and divide each term by u⁴.