what are the zeroes inclucing irrational of this polynomial? 3x^4 - 14x^2 - 5

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If we consider the transformation `z=x^2` , then the polynomial is:

`3x^4-14x^2-5`

`=3z^2-14z-5`  factor this using decomposition

`=3z^2-15z+z-5`

`=3z(z-5)+(z-5)`

`=(3z+1)(z-5)`  now goe back to x

`=(3x^2+1)(x^2-5)`   the left term has complex roots, the right irrational

`=(3x^2+1)(x-sqrt5)(x+sqrt5)`

This means that the real zeros are `sqrt5` and `-sqrt5` .

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If we consider the transformation `z=x^2` , then the polynomial is:

`3x^4-14x^2-5`

`=3z^2-14z-5`  factor this using decomposition

`=3z^2-15z+z-5`

`=3z(z-5)+(z-5)`

`=(3z+1)(z-5)`  now goe back to x

`=(3x^2+1)(x^2-5)`   the left term has complex roots, the right irrational

`=(3x^2+1)(x-sqrt5)(x+sqrt5)`

This means that the real zeros are `sqrt5` and `-sqrt5` .

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