Simplify \( [1 ÷ (x^2 + 3x + 2)] + [1 ÷ (x^2 + 5x + 6)] \)
A. \( \frac{2}{(x + 1)^2} \)
B. \( \frac{2}{(x + 1)(x + 2} \)
C. \( \frac{2}{(x + 1)(x + 2} \)
D. \( \frac{2}{(x + 1)(x + 3} \)
Correct Answer: D
Explanation
\( [1 ÷ (x^2 + 3x + 2)] + [1 ÷ (x^2 + 5x + 6)] \)
= \( 1 ÷ (x^2 + 3x + 2) + [1 ÷ (x^2 +5x + 6)]\)
= \( [1 ÷ ((x^2 + x) + (2x + 2) )] + [1 ÷ ((x^2 + 3x) + (2x + 6) )] \)
= [1 ÷ (x(x + 2) + 2(x +1))] + [1 ÷ (x(x + 3) +2(x + 3) )]
= [1 ÷ (x + 1)(x + 2)] + [1 ÷ ((x + 3) + (x + 2))]
=((x + 3) + (x + 1)) ÷ (x + 1)(x + 2)(x + 3)
Using the L.C.M
=((x + x + 3 + 1)) ÷ (x + 1)(x + 2)(x + 3)
=(2x+4)/(x+1)(x+2)(x+3) =2(x+2)/(x+1)(x+2)(x+3)
= \( \frac{2 }{(x + 1)(x + 3)} \)
