If frac{x^3}{(2x - 1)(x + 2)(x - 3)} = p + fr | vedclass

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(D) Given the equation: $\frac{x^3}{(2x - 1)(x + 2)(x - 3)} = p + \frac{q}{2x - 1} + \frac{r}{x + 2} + \frac{s}{x - 3}$.Multiply both sides by $(2x -

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$6q - 3r + 2s = 3$

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(D) Given the equation: $\frac{x^3}{(2x - 1)(x + 2)(x - 3)} = p + \frac{q}{2x - 1} + \frac{r}{x + 2} + \frac{s}{x - 3}$.Multiply both sides by $(2x - 1)(x + 2)(x - 3)$:$x^3 = p(2x - 1)(x + 2)(x - 3) + q(x + 2)(x - 3) + r(2x - 1)(x - 3) + s(2x - 1)(x + 2)$.Expanding the term with $p$: $p(2x - 1)(x^2 - x - 6) = p(2x^3 - 2x^2 - 12x - x^2 + x + 6) = p(2x^3 - 3x^2 - 11x + 6)$.Equating the coefficient of $x^3$ on both sides: $1 = 2p \Rightarrow p = \frac{1}{2}$.Now,to find the constant term,set $x = 0$ in the equation:$0 = p(-1)(2)(-3) + q(2)(-3) + r(-1)(-3) + s(-1)(2)$.$0 = 6p - 6q + 3r - 2s$.Substituting $p = \frac{1}{2}$:$0 = 6(\frac{1}{2}) - 6q + 3r - 2s$.$0 = 3 - 6q + 3r - 2s$.$6q - 3r + 2s = 3$.

If $\frac{x^3}{(2x - 1)(x + 2)(x - 3)} = p + \frac{q}{2x - 1} + \frac{r}{x + 2} + \frac{s}{x - 3}$,then which of the following is true?

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