Theory question 103
PolynomialsGiven the function f(x) = (2x3 − x)/(x2 − 1), x ≠ ±1, express :
(a) f(x) in the form g(x) + h(x); (b) h(x) in partial fractions
Show the model answer
Model answer
(a) 2x³ − x = 2x(x² − 1) + x, so f(x) = 2x + x/(x² − 1): g(x) = 2x, h(x) = x/(x² − 1).
(b) x/((x − 1)(x + 1)) = A/(x + 1) + B/(x − 1); x = A(x − 1) + B(x + 1).
x = 1: B = ½; x = −1: A = ½.
h(x) = 1/(2(x + 1)) + 1/(2(x − 1)).
Explanation
Divide first (the top has a higher degree), then split the proper fraction.