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To simplify the expression (3x² – x – 2) / (x² – 1), we begin by factoring both the numerator and the denominator. The numerator, 3x² – x – 2, can be factored by looking for two numbers that multiply to give the product of the leading coefficient (3) and the constant term (-2), which is -6, while also summing to the middle coefficient (-1). The numbers -3 and 2 satisfy these conditions, allowing us to rewrite the expression as 3x² – 3x + 2x – 2. This can be grouped and factored further, resulting in (3x + 2)(x – 1).
Next, we turn our attention to the denominator, x² – 1, which is a difference of squares. This can be factored into (x – 1)(x + 1). With both the numerator and denominator factored, we can rewrite the original expression as [(3x + 2)(x – 1)] / [(x – 1)(x + 1)].
At this stage, we observe that the term (x – 1) appears in both the numerator and the denominator, allowing us to cancel it out, provided that x is not equal to 1. This leads us to the simplified expression of (3x + 2) / (x + 1). Thus, the final result of the simplification process is (3x + 2) / (x + 1), valid for all x except x = 1.
