Solution:

To find the remaining zeroes of the given polynomial equation, we can use the fact that if two of its zeroes are 3 and 1, then the polynomial must be divisible by (x-3) and (x-1).

Let's factorize the polynomial equation using these two known zeroes:

x^4 + 7x - 17x^2 - 17x = 6

Rearranging the terms, we get:

x^4 - 17x^2 + 7x - 17x = 6

x(x^3 - 17x + 7 - 17) = 6

x(x^3 - 17x - 10) = 6

Since (x-3) and (x-1) are factors of the polynomial, we can write:

x^3 - 17x - 10 = 6/(x-3)(x-1)

Now, let's find the remaining zeroes using the factor theorem:

1. Synthetic Division:
Using synthetic division, we can divide the polynomial (x^3 - 17x - 10) by (x-3).

Using synthetic division, we get:

3 | 1 0 -17 -10
| 3 9 -24
-----------------------
1 3 -8 -34

The quotient is x^2 + 3x - 8 with a remainder of -34.

Hence, we have reduced the polynomial to:

(x-3)(x^2 + 3x - 8) = 6/(x-3)(x-1)

2. Quadratic Equation:
Now, let's solve the quadratic equation x^2 + 3x - 8 = 6/(x-1) to find the other two zeroes.

Multiplying both sides by (x-1), we get:

(x-1)(x^2 + 3x - 8) = 6

Expanding the equation, we have:

x^3 + 3x^2 - 8x - x^2 - 3x + 8 = 6

Simplifying further, we get:

x^3 + 2x^2 - 11x + 2 = 0

Now, we have reduced the polynomial to a cubic equation. To solve this equation and find the remaining zeroes, we can use numerical methods such as the Newton-Raphson method or graphical methods like plotting the graph and finding the roots.

3. Additional Methods:
Other methods like factoring, completing the square, or using the cubic formula can also be used to find the remaining zeroes of the cubic equation.

In conclusion, by using the known zeroes and applying the factor theorem, we were able to reduce the given polynomial equation and find a cubic equation. The remaining zeroes can be found by solving the cubic equation using numerical or graphical methods.