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Solve the equation x⁴ - 2x³ + 3x² - 2x + 1 = 0.

1. x = 1
2. x = -1
3. x = (1 +- sqrt5)/2
4. x = (3 +- sqrt5)/2

Correct answer: x = (1 +- sqrt5)/2

Solution

The coefficients are palindromic, so divide by x² and substitute y = x + 1/x. This reduces to a quadratic in y. Solving gives y, then solving x + 1/x = y for each y produces the roots. The real roots are x = (1 +- sqrt5)/2 (the others are complex).

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