Find all the zeros of (2x^{4} ‒ 3x^{3} ‒ 5x^{2} + 9x ‒ 3), it being given that two of its zeros are √3 and – √3.

Let us assume f (x) = 2x^{4} - 3x^{3} ‒ 5x^{2} + 9x - 3

As and – are the zeros of the given polynomial therefore each one of (x - ) and (x + ) is a factor of f (x)

Consequently, (x – ) (x + ) = (x^{2} – 3) is a factor of f (x)

Now, on dividing f (x) by (x^{2} – 3) we get:

f (x) = 0

2x^{2} - 3x^{2} – 5x^{2} + 9x - 3 = 0

(x^{2} – 3) (2x^{2} – 3x + 1) = 0

(x^{2} - 3) (2x^{2} – 2x – x + 1) (2x – 1) (x - 1) = 0

(x – ) (x + ) (2x – 1) (x - 1) = 0

∴ x = or x = - or x = or x = 1

Hence, all the zeros of the given polynomial are , -, and 1

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