Question 131842
You have it right.  Just look at your zeros, {{{1-sqrt(13)}}} and {{{1+sqrt(13)}}} as if they were just numbers (which they are actually), and do what you would do with a number.


a is a root of the polynomial equation if and only if {{{x-a}}} is a factor of the polynomial, so for your problem the factors are:

{{{x-(1-sqrt(13))}}} and {{{x-(1+sqrt(13))}}}, so:


{{{y=(x-1+sqrt(13))(x-1-sqrt(13))}}}


Now you have to extend the FOIL process somewhat to multiply what are actually trinomials, but if you follow a pattern, it will all work out.


{{{y=x^2-x-x*sqrt(13)-x+1+sqrt(13)+x*sqrt(13)-sqrt(13)-13}}}


Now collect terms:
{{{y=x^2-2x-12}}}.  Done.


To check the answer, you can complete the square on the resultant polynomial:
Set the polynomial equal to 0:
{{{x^2-2x-12=0}}}


Put the constant on the right:
{{{x^2-2x=12}}}


Divide the first degree term coefficient by 2 and square the result, adding the result to both sides:
{{{x^2-2x+1=13}}}


{{{(x-1)^2=13}}}


{{{x-1=sqrt(13)}}} or {{{x-1=-sqrt(13)}}}


{{{x=1+sqrt(13)}}} or {{{x=1-sqrt(13)}}}.  Answer checks.