We are asked to determine a polynomial that has the following roots:
[tex]1+\sqrt[]{2},2-i[/tex]As roots. We will use a polynomial of degree 4 that has the roots:
[tex]1\pm\sqrt[]{2},2\pm i[/tex]This means that the factors of the polynomials are:
[tex](x-(1+\sqrt[]{2})(x-(1-\sqrt[]{2}))(x-(2-i))(x-(2+i))[/tex]Now we must expand the given factor in order to get a polynomial of rational coefficients. First, we will take the two first products:
[tex](x-(1+\sqrt[]{2})(x-(1-\sqrt[]{2}))[/tex]Now we will reassociate terms inside each parenthesis:
[tex]((x-1)-\sqrt[]{2})((x-1)+\sqrt[]{2}))[/tex]Now we apply the distributive law using the associated terms:
[tex](x-1)^2+\sqrt[]{2}(x-1)-\sqrt[]{2}(x-1)-(\sqrt[]{2})^2[/tex]Simplifying:
[tex](x-1)^2-2[/tex]Therefore, the first two products can be replaced by the term we just found:
[tex]((x-1)^2-2)(x-(2-i))(x-(2+i))[/tex]Now we take the third and fourth products:
[tex](x-(2-i))(x-(2+i))[/tex]Now we reassociate the terms:
[tex]((x-2)+i))((x-2)-i))[/tex]Now we apply the distributive law:
[tex](x-2)^2+i(x-2)-i(x-2)-(i)^2[/tex]Simplifying we get:
[tex](x-2)^2+1[/tex]Now we can replace this for the third and fourth products:
[tex]((x-1)^2-2)((x-2)^2+1)[/tex]Now we solve the squares in each parenthesis:
[tex](x^2-2x+1-2)(x^2-4x+4+1)[/tex]Adding like terms:
[tex](x^2-2x-1)(x^2-4x+5)[/tex]Now we apply the distributive property:
[tex]x^4-4x^3+5x^2-2x^3+8x^2-10x-x^2+4x-5[/tex]Adding like terms we get:
[tex]x^4-6x^3+12x^2-6x-5[/tex]And thus we get the desired polynomial.