Write a polynomial with these root

Question

1. 2-i, √5
2. √3, 2i

Answer 1

For real coefficients, 2+i is also a solution;

for RATIONAL coefficients, -sqrt(5) is also a solution;

Since it is not specified that the polynomial must have rational coefficients,

a cubic polynomial is sufficient with factors (x-2 + i)(x-2 - i)(x - sqrt(5))

[ (x-2)^2 - (-1)][ x - sqrt(5)]

[ x^2 - 4x + 4 +1](x - sqrt(5)]

[ x^2 - 4x + 5](x - sqrt(5))

If you want RATIONAL coefficient, then as stated, you must also factor in (x + sqrt(5)),

which FOILs out to (x^2 - 5) with it's conjugate.

LIkewise for your second problem, the BARE MINIMUM for REAL coefficients is

(x - 2i)(x+2i)(x-sqrt(3)) = (x^2 + 4)(x - sqrt(3))

For RATIONAL coefficients, again, you will need to factor in (x + sqrt(3)) , which

FOILS out to x^2 - 3 with it's conjugate.