Find all the zeroes of the polynomial x⁴ + 2x³ – 17x² – 4x + 30, given that two of its zeroes are 3 and – 5.

Given zeroes are 1 and 2

So, (x – 3)and [x – (-5)] are the factors of x⁴ + 2x³ – 17x² – 4x + 30

⇒ (x – 3)(x + 5)

= x² + 5x – 3x – 15

= x² + 2x – 15 is a factor of given polynomial.

Consequently, x² + 2x – 15 is also a factor of the given polynomial.

Now, let us divide x⁴ + 2x³ – 17x² – 4x + 30 by x² + 2x – 15

The division process is

Here, quotient = x² – 2

= (x – 2)(x + 2)

So, the zeroes are -2 and 2

Hence, all the zeroes of the given polynomial are -2, 2, 3 and 5.