Question

If \[ \frac{x + 2}{x^2 - 3} \text{ is one of the partial fractions of } \frac{3x^3 - x^2 - 2x + 17}{x^4 + x^2 - 12}, \text{ then the other partial fraction of it is:} \]

• A. \( \frac{2x + 3}{x^2 - 4} \)
• B. \( \frac{3x + 2}{x^2 + 4} \)
• C. \( \frac{2x - 3}{x^2 + 4} \)
• D. \( \frac{3x - 2}{x^2 - 4} \)

Hint:

For partial fraction decomposition, ensure that the denominator is factored correctly and combine the numerators over the common denominator.

Answer 1

The Correct Option is C

We are given the equation: \[ \frac{3x^3 - x^2 - 2x + 17}{x^4 + x^2 - 12} = \frac{x + 2}{x^2 - 3} + \frac{2x - 3}{x^2 + 4} \] The first step is to factorize the denominator of the right-hand side expression. Notice that the denominator \( x^4 + x^2 - 12 \) can be factored as: \[ x^4 + x^2 - 12 = (x^2 - 3)(x^2 + 4) \] Now, rewrite the partial fractions with this common denominator: \[ \frac{x + 2}{x^2 - 3} = \frac{(x + 2)(x^2 + 4)}{(x^2 - 3)(x^2 + 4)} \] \[ \frac{2x - 3}{x^2 + 4} = \frac{(2x - 3)(x^2 - 3)}{(x^2 - 3)(x^2 + 4)} \] Now, add these two fractions: \[ \frac{(x + 2)(x^2 + 4) + (2x - 3)(x^2 - 3)}{(x^2 - 3)(x^2 + 4)} \] Simplify the numerator: \[ (x + 2)(x^2 + 4) = x^3 + 4x + 2x^2 + 8 \] \[ (2x - 3)(x^2 - 3) = 2x^3 - 6x - 3x^2 + 9 \] Add the two expressions: \[ x^3 + 4x + 2x^2 + 8 + 2x^3 - 6x - 3x^2 + 9 = 3x^3 - x^2 - 2x + 17 \] Thus, the numerator simplifies to \( 3x^3 - x^2 - 2x + 17 \), which matches the numerator on the left-hand side of the equation. Therefore, the correct answer is \( \frac{2x - 3}{x^2 + 4} \), corresponding to option (3).