Question

okay so im trying to do test corrections and i have "a fourth-degree polynomial with integer coefficients has roots at 1 and 3+ the square root of 5. which number cannot also be a root of this polynomial?"

Answer 1

Correction:
If 3+sqrt(5) is a root, and the polynomial has integer coefficients, then the conjugate (3-sqrt(5)) must be one of the remaining roots. That makes two roots out of 4.

Now 1 is a root, so the remaining root cannot be irrational, otherwise the coefficient cannot be integral.

So any irrational (other than 3+/- sqrt(5)) cannot be one of the remaining roots. Example: 2+sqrt(3), 5+sqrt(7), etc.