If is a zero of a polynomial function, then is a factor of the polynomial.
Complex zeros and irrational zeros come in conjugate pairs. That is if is a zero, then is also a zero. Also if where is rational and is irrational is a zero, then is also a zero.
Therefore four of the five factors of your desired polynomial function are:
Since is not irrational, there is no guarantee that the conjugate is a zero. There is no way to determine the fifth zero from the information given and therefore there is no way to determine a unique 5th degree polynomial.
John
My calculator said it, I believe it, that settles it
