To find the zeroes of the given polynomial [tex]$3x^4 + 6x^3 - 2x^2 - 10x + 5$[/tex], we begin by analyzing the given information, which states that [tex]$\sqrt{\frac{5}{3}}$[/tex] and [tex]$-\sqrt{\frac{5}{3}}$[/tex] are some of the zeroes of the polynomial.
Given:
[tex]\[ \sqrt{\frac{5}{3}} \approx 1.2909944487358056 \][/tex]
[tex]\[ -\sqrt{\frac{5}{3}} \approx -1.2909944487358056 \][/tex]
These values are the zeroes of the polynomial. The numerical approximations of these zeroes are:
[tex]\[ x_1 \approx 1.2909944487358056 \][/tex]
[tex]\[ x_2 \approx -1.2909944487358056 \][/tex]
Hence, the zeroes of the polynomial [tex]$3x^4 + 6x^3 - 2x^2 - 10x + 5$[/tex] are:
[tex]\[ \boxed{1.2909944487358056 \text{ and } -1.2909944487358056} \][/tex]
