Please, give me a hint to find a polynomial of 4th order. I only know 2 roots: 1-square root 2 and 2-i.

We'll recall the fact that complex roots and irrational
roots come in pairs. Therefore, if x1 = 1 - sqrt2 is a root of the polynomial, then x2 =
1 + sqrt 2 is also a root of the polynomial. Also, if x3 = 2 - i is another root of
polynomial, then it's conjugate, x4 = 2 + i, is a root of the polynomial,
too.

We'll write the 4th order polynomial as a product of
linear factors:

P(x) = (x - x1)(x - x2)(x - x3)(x -
x4)

P(x) = (x - 1 + sqrt2)(x - 1 - sqrt2)(x - 2 + i)(x - 2
- i)

P(x) = [(x-1)^2 - 2]*[(x-2)^2 -
i^2]

But i^2 = -1. Expanding the binomials inside brackets,
we'll get:

P(x) = (x^2 - 2x + 1 - 2)*(x^2 - 4x + 4 +
1)

Combining like terms inside brackets, we'll
get:

P(x) = (x^2 - 2x - 1)*(x^2 - 4x +
5)

We'll remove the
brackets:

P(x) = x^4 - 4x^3 + 5x^2 - 2x^3 + 8x^2 - 10x -
x^2 + 4x - 5

Combining like terms, we'll
get:

P(x) = x^4 - 6x^3 + 12x^2 - 6x -
5

The 4th order requested polynomial is: P(x)
= x^4 - 6x^3 + 12x^2 - 6x - 5.