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Solve (3x⁴ + x² - 2x - 3)/(3x⁴ - x² + 2x + 3) = (5x⁴ + 2x² - 7x + 3)/(5x⁴ - 2x² + 7x - 3).

1. x = 1 or x = -1
2. x = 1 only
3. x = 0 or x = 1
4. x = ±3

Correct answer: x = 1 or x = -1

Solution

Each fraction has numerator and denominator differing only in the signs of the lower-degree terms. Using componendo-dividendo, (N+D)/(N-D) for the left side = (6x⁴)/(2x² - 4x - 6) and for the right = (10x⁴)/(4x² - 14x + 6). Setting these equal and simplifying leads to a polynomial whose real roots are x = 1 and x = -1 (with x = 1 and x = -1 satisfying the original after checking denominators are nonzero).

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