Express (frac{x^{2} + x + 4}{(1 – x)(x^{2} + 1)}) in partial fractions.
-
A.
(frac{x^{2}}{x^{2} + 1} + frac{x + 4}{1 – x}) -
B.
(frac{3}{1 – x} + frac{2x + 1}{x^{2} + 1}) -
C.
(frac{x^{2}}{1 – x} + frac{x + 4}{x^{2} + 1}) -
D.
(frac{3}{1 – x} + frac{2x + 2}{x^{2} + 1})
Correct Answer: Option B
Explanation
(frac{x^{2} + x + 4}{(1 – x)(x^{2} + 1)} = frac{A}{1 – x} + frac{Bx + C}{x^{2} + 1})
= (frac{A(x^{2} + 1) + (Bx + C)(1 – x)}{(1 – x)(x^{2} + 1)})
(implies x^{2} + x + 4 = A(x^{2} + 1) + (Bx + C)(1 – x))
(x^{2} + x + 4 = Ax^{2} + A + Bx – Bx^{2} – Cx + C)
(implies (A – B)x^{2} = x^{2}; A – B = 1 …… (i))
((B – C)x = x; B – C = 1 ….. (ii))
(A + C = 4 …… (iii))
Solving the above simultaneous equations by any of the known methods, we get
(A = 3, B = 2, C = 1)
(therefore frac{x^{2} + x + 4}{(1 – x)(x^{2} + 1)} = frac{3}{1 – x} + frac{2x + 1}{x^{2} + 1})