frac{x^2 + 13x + 15}{(2x + 3)(x + 3)^2} = | vedclass

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(A) Let $\frac{x^2 + 13x + 15}{(2x + 3)(x + 3)^2} = \frac{A}{2x + 3} + \frac{B}{x + 3} + \frac{C}{(x + 3)^2}$.Multiplying both sides by $(2x + 3)(x +

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Let $\frac{x^2 + 13x + 15}{(2x + 3)(x + 3)^2} = \frac{A}{2x + 3} + \frac{B}{x + 3} + \frac{C}{(x + 3)^2}$.Multiplying both sides by $(2x + 3)(x + 3)^2$,we get:$x^2 + 13x + 15 = A(x + 3)^2 + B(2x + 3)(x + 3) + C(2x + 3)$.For $x = -3$,$(-3)^2 + 13(-3) + 15 = C(2(-3) + 3)$ $\Rightarrow 9 - 39 + 15 = -3C$ $\Rightarrow -15 = -3C$ $\Rightarrow C = 5$.For $x = -\frac{3}{2}$,$(-\frac{3}{2})^2 + 13(-\frac{3}{2}) + 15 = A(-\frac{3}{2} + 3)^2$ $\Rightarrow \frac{9}{4} - \frac{39}{2} + 15 = A(\frac{3}{2})^2$ $\Rightarrow \frac{9 - 78 + 60}{4} = \frac{9}{4}A$ $\Rightarrow -9 = 9A$ $\Rightarrow A = -1$.Equating the coefficients of $x^2$ on both sides:$1 = A + 2B$ $\Rightarrow 1 = -1 + 2B$ $\Rightarrow 2B = 2$ $\Rightarrow B = 1$.Substituting the values of $A, B, C$ back into the partial fraction form:$\frac{1}{x + 3} - \frac{1}{2x + 3} + \frac{5}{(x + 3)^2}$.

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

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