Question

$\frac{x^{2} + 13x + 15}{(2x + 3)(x + 3)^{2}} =$

• A. $\frac{1}{x + 3} - \frac{1}{2x + 3} + \frac{5}{(x + 3)^{2}}$
• B. $\frac{1}{2x + 3} - \frac{1}{x + 3} + \frac{5}{(x + 3)^{2}}$
• C. $\frac{1}{2x + 3} + \frac{1}{x + 3} - \frac{5}{(x + 3)^{2}}$
• D. $\frac{1}{2x + 3} - \frac{1}{x + 3} - \frac{5}{(x + 3)^{2}}$

Answer 1

Answer: (A)

$\frac{1}{x + 3} - \frac{1}{2x + 3} + \frac{5}{(x + 3)^{2}}$

Answer 2

$\frac{x^{2} + 13x + 15}{(2x + 3)(x + 3)^{2}} = \frac{A}{2x + 3} + \frac{B}{x + 3} + \frac{C}{(x + 3)^{2}}$

⇒ $x^{2} + 13x + 15 = A(x + 3)^{2} + B(2x + 3)(x + 3) + C(2x + 3)$

For $x = - 3,C = 5$ and for $x = - \frac{3}{2};A = - 1$

Equating coefficient of $x^{2}$

$1 = A + 2B \Rightarrow B = \frac{1 - A}{2} = 1$

$\therefore$Given expression = $\frac{1}{x + 3} - \frac{1}{2x + 3} + \frac{5}{(x + 3)^{2}}$.