What is a quartic polynomial equation with rational coefficients that has roots at i and 2i

Question

A. X^4-5x^2-4

B. X^4-5x^2+4

C. X^4+5x^2+4

D. X^4+5x^2-4

Answer 1

I^4 = 1

i^2 = -1

plug i into A, you get 1+5 -4 = 2

plug i into B, you get 1+5+4=9

plug i into C you get 1-5+4 = 0

Bingo, that's one possible solution, C

plug i into D you get 1-5-4=-8

Only C has i for a root

If this were a time pressure test, you'd check C and move on to the next question. It would be at least tied for the best answer even if 2i worked for either A, B and D.

But if you have time you also try to plug in the other root, 2i, and see if it works with C

2i^4 = 16

2i^2 = -4

For C, plug them in, you get 16-20+4 = 0

C is the correct answer.

You could check and see if 2i worked for A, B and C, but since i doesn't work for them, they are incorrect