Express (frac{5x – 12}{(x – 2)(x – 3)}) in partial fractions
-
A.
(frac{2}{x + 2} – frac{3}{x – 3}) -
B.
(frac{2}{x – 2} + frac{3}{x – 3}) -
C.
(frac{2}{x – 3} – frac{3}{x – 2}) -
D.
(frac{5}{x – 3} – frac{4}{x – 2})
Correct Answer: Option B
Explanation
(frac{5x – 12}{(x – 2)(x – 3)} = frac{A}{x – 2} + frac{B}{x – 3})
= (frac{A(x – 3) + B(x – 2)}{(x – 2)(x – 3)})
(implies 5x – 12 = Ax – 3A + Bx – 2B)
(A + B = 5 … (i))
(-(3A + 2B) = -12 implies 3A + 2B = 12 … (ii))
From (i), (A = 5 – B)
(3(5 – B) + 2B = 12)
(15 – 3B + 2B = 12 implies B = 3)
(A + 3 = 5 implies A = 2)
(frac{5x – 12}{(x – 2)(x – 3)} = frac{2}{x – 2} + frac{3}{x – 3})