We can use the quadratic formula to find the zeroes of the given polynomial. The quadratic formula states that for a quadratic equation of the form ax² + bx + c = 0, the solutions for x are given by:
x = (-b ± sqrt(b² - 4ac)) / 2a
Comparing this to the given polynomial, we see that a = 3, b = -2√3, and c = 1. Substituting these values into the quadratic formula, we get:
x = [(-(-2√3)) ± sqrt((-2√3)² - 4(3)(1))] / 2(3)
x = [(2√3) ± sqrt(12 - 4)] / 6
x = [(2√3) ± sqrt(8)] / 6
x = [(2√3) ± 2√2] / 6
x = (√3 ± √2) / 3
Therefore, the zeroes of the polynomial 3x² - 2√3x + 1 are (√3 + √2) / 3 and (√3 - √2) / 3.