If a polynomial function f(x) has roots 1+square root of 2 and-3, what must be a factor of f(x)?

To find the factor a a polynomial from its roots, we are going to seat each one of the roots equal to[tex]x[/tex], and then we are going to factor backwards.

We know for our problem that one of the roots of our polynomial is -3, so lets set -3 equal to[tex]x[/tex] and factor backwards:
[tex]x=-3[/tex]
[tex]x+3=0[/tex]
[tex](x+3)[/tex] is a factor of our polynomial.

We also know that another root of our polynomial is [tex]1+ \sqrt{2} [/tex], so lets set [tex]1+ \sqrt{2} [/tex] equal to [tex]x[/tex] and factor backwards:
[tex]x=1+ \sqrt{2} [/tex]
[tex]x-1= \sqrt{2} [/tex]
[tex]x-1- \sqrt{2}=0 [/tex]
[tex](x-(1+ \sqrt{2})=0 [/tex]
([tex](x-(1+ \sqrt{2} ))[/tex] is a factor of our polynomial.

We can conclude that there is no correct answer in your given choices.