Find a polynomial of degree 4 with integer coefficients that has -1 as its only real zero and i as a zero.

Answer 1

(x+1)^2(x-i)(x+i) =

(x^2+2x+1)(x^2+1) =

x^4 + 2x^3 +2x^2 +2x + 1

imaginary zeroes come in conjugate pairs. if i is a zero so is -i

then stick an x in front of them to get (x-i)(x+i) = x^2+1

IF -1 is the only real zero, then it would have to have multiplicity 2 if you wanted a 4th degree polynomial

(x+1)(x+1) = (x+1)^2 = x^2 + 2x + 1

multiply those two together (x^2+2x+1)(x^2+1) to get the 4th degree polynomial