Find all the zero

Let p(x) = 2x⁴ – 3x³ – 3x² + 6x – 2

Now, 1 and 1/2 are the zeroes of the polynomial. So, by factor theorem we know that (x – 1) and (x – 1/2) will completely divide p(x)

(x – 1)(2x – 1) is a factor of p(x)

2x² – 3x + 1 is a factor of p(x).

Dividing p(x) by 2x² – 3x + 1, we get,

Therefore,

2x⁴ – 3x³ – 3x² + 6x – 2 = (2x² – 3x + 1)(x² – 2)

Now, for finding the zeroes,

Putting (2x² – 3x + 1)(x² – 2) = 0

x² – 2 = 0

(x + √2)(x - √2) = 0

x = -√2 or x = √2

Hence, the zeroes of the polynomial 2x⁴ – 3x³ – 3x² + 6x – 2 are 1, 1/2 , √2 and -√2.