Question

what is the difference?
\\(\frac{2x + 5}{x^2 - 3x} - \frac{3x + 5}{x^3 - 9x} - \frac{x + 1}{x^2 - 9}\\)
\\(\circ \frac{(x + 5)(x + 2)}{x^3 - 9x}\\)
\\(\circ \frac{(x + 5)(x + 4)}{x^3 - 9x}\\)
\\(\circ \frac{-2x + 11}{x^3 - 12x - 9}\\)
\\(\circ \frac{3(x + 2)}{x^2 - 3x}\\)

Explanation:

Step1: Factor Denominators

Factor each denominator:

• \(x^2 - 3x = x(x - 3)\)
• \(x^3 - 9x = x(x^2 - 9) = x(x - 3)(x + 3)\)
• \(x^2 - 9 = (x - 3)(x + 3)\)

Step2: Find Common Denominator

The least common denominator (LCD) is \(x(x - 3)(x + 3)\) (which is \(x^3 - 9x\)).

Step3: Rewrite Fractions with LCD

• \(\frac{2x + 5}{x(x - 3)} = \frac{(2x + 5)(x + 3)}{x(x - 3)(x + 3)}\)
• \(\frac{3x + 5}{x(x - 3)(x + 3)}\) (already has LCD)
• \(\frac{x + 1}{(x - 3)(x + 3)} = \frac{x(x + 1)}{x(x - 3)(x + 3)}\)

Step4: Subtract the Fractions

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Answer:

\(\frac{(x + 5)(x + 2)}{x^3 - 9x}\) (the first option)