Word Count : 499

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We want to express the given rational function as a sum of partial fractions: \[ \frac{x - 1}{x^3 - x^2 - 2x} \] ### Step 1: Factor the denominator The denominator is: \[ x^3 - x^2 - 2x \] Factor out the common term \( x \): \[ x(x^2 - x - 2) \] Now, factor the quadratic \( x^2 - x - 2 \): \[ x^2 - x - 2 = (x - 2)(x + 1) \] Thus, the fully factored denominator is: \[ x(x - 2)(x + 1) \] ### Step 2: Set up partial fraction decomposition We express the fraction as: \[ \frac{x - 1}{x(x - 2)(x + 1)} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 1} \] ### ___ ______ ___ _____ __________ _______ _______ ________ ___ _______ _______ ____.
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