Given Equation: 3x^4 - 15x^3 + 13x^2 - 25x - 30 = 0

Given Zeroes: √5/3

To find the remaining zeroes of the given equation, we can use the fact that if α is a zero of a polynomial, then (x - α) is a factor of the polynomial.

Step 1: Divide the given equation by (x - α)

Using the given zero √5/3, we can divide the given equation by (x - √5/3) using long division or synthetic division.

_________
x - √5/3 | 3x^4 - 15x^3 + 13x^2 - 25x - 30
- (3x^4 - √5x^3)
________________
(√5/3)x^3 + 13x^2 - 25x - 30
- (√5/3)x^3 + (√5/3)x^2
_______________________
13x^2 - 25x - 30
- 13x^2 + (√5/3)x
_____________________
- 25x - (√5/3)x - 30
+ 25x - (5/3)
______________________
-(√5/3)x - (5/3)

The quotient obtained after division is (√5/3)x^3 + (√5/3)x^2 - (√5/3)x - (5/3).

Step 2: Factorize the quotient

To find the remaining zeroes, we need to factorize the quotient (√5/3)x^3 + (√5/3)x^2 - (√5/3)x - (5/3).

Factoring out (√5/3)x from the first two terms and - (5/3) from the last two terms, we get:

(√5/3)x^2((√5/3)x + 1) - (5/3)((√5/3)x + 1)

Notice that (√5/3)x + 1 is common in both terms. We can factor this out:

(√5/3)x^2((√5/3)x + 1) - (5/3)((√5/3)x + 1) = ((√5/3)x + 1)((√5/3)x^2 - (5/3))

Step 3: Set each factor equal to zero and solve for x

Setting (√5/3)x + 1 = 0, we can solve for x:

(√5/3)x = -1
x = -√5/3

Setting (√5/3)x^2 - (5/3) = 0, we can solve for x:

(√5/3)x^2 = 5/3
x^2 = 5/3 * (3/√5)
x^2 = 5/√5
x^2 = √5
x = ±