Roots of polynomial $x^3-3\sqrt 5x^2+13x-3\sqrt 5$ given the factor $x-\sqrt 5$ 
Given that $x-\sqrt 5 $ is a factor of the cubic polynomial
  $x^3-3\sqrt 5x^2+13x-3\sqrt 5$, find all the values of
  the polynomial

After the long division method I get $x^2-2\sqrt 5x+3$.
Now how to split the middle term to find the all zeros of the polynomial?

As an alternate method, you can use Vieta's formulas: if $a, b, c$ are the three roots, you have
$$\begin{eqnarray}
(i) & a+b+c&=&3\sqrt{5}\\
(ii) & ab+bc+ca&=&13\\
(iii) & abc&=&3\sqrt{5}\end{eqnarray}$$
One of the roots, say $c$, is $\sqrt{5}$, so


*

*from $(i)$, $a+b=2\sqrt{5}$,

*from $(iii)$, $ab=3$,


So
$$(a-b)^2=(a+b)^2-4ab=20-12=8$$
Then from $a+b=2\sqrt{5}$ and $a-b=2\sqrt{2}$, you have immadiately
$$a=\sqrt{5}+\sqrt{2}\\
b=\sqrt{5}-\sqrt{2}$$
