Find a polynomial of degree four with integer coefficients and no real roots such that i/2 is a root. Solution The polynomial ( x - i) ( x + i) ( x - i/2) ( x + i/2 ) has no real roots.Its roots are i/2, - i/2, i and - i. On expansion, the polynomial is ( x2 - i2 ) ( x2 - i2/ 4) or, ( x2 + 1)(x2+ 1/4) 0r, x4 + (5/4) x2 + 1/4 Apparently, the other roots here are, -i/2 , i and -i..

1. Find a polynomial of degree four with integer coefficients and no real roots such that i/2 is a
root.
Solution
The polynomial ( x - i) ( x + i) ( x - i/2) ( x + i/2 ) has no real roots.Its roots are i/2, - i/2, i and - i.
On expansion, the polynomial is ( x2 - i2 ) ( x2 - i2/ 4) or, ( x2 + 1)(x2+ 1/4) 0r, x4 + (5/4) x2 +
1/4 Apparently, the other roots here are, -i/2 , i and -i.