<s>[INST] Finding polynomial function's zero values not native English speaker so I may get some terms wrong and so on.
On to the question:
I have as an assignment to find a polynomial function $f(x)$ with the coefficients $a$, $b$ and $c$ (which are all integers) which has one root at $x = \sqrt{a} + \sqrt{b} + \sqrt{c}$.
I've done this with $f(x) = 0$ where $x = \sqrt{a} + \sqrt{b}$ through an iterative method which looks like this (forgive me for my pseudo code):

  
*
  
*var x = sqrt(a) + sqrt(b)
  
*Set x to x multiplied by the conjugate of x
  
*Repeat from step 2 until all square roots are gone
  

The full calculation looks like this (the exponent signs disappeared, sorry about that):
$(x-(\sqrt{a}+\sqrt{b}))(x+(\sqrt{a}+\sqrt{b}))
= x^2-(\sqrt{a} +\sqrt{b})^2 \\
= x^2-(a+2\sqrt{a}\sqrt{b}  +b) \\
= x^2-a-b-2\sqrt{a}\sqrt{b}  $
$((x^2-a-b)-(2\sqrt{a}\sqrt{b} ))(( x^2-a-b)+(2\sqrt{a}\sqrt{b}  )) 
= x^4-2ax^2-2bx^2+2ab+a^2+b^2-4ab \\
= x^4-2ax^2-2bx^2-2ab+a^2+b^2 \\
= x^4-2x^2(a+b)-2ab+a^2+b^2$
$p(x) = x^4-2x^2(a+b)-2ab+a^2+b^2$
And then I used the factor theorem to calculate the remaining roots, which gave the following results:
$x = (\sqrt{a} +\sqrt{b} )$, $x = -(\sqrt{a}  +\sqrt{b} )$, $x = (\sqrt{a} -\sqrt{b} )$ and $x = -(\sqrt{a} - \sqrt{b})$.
When I try the same method on  $x = \sqrt{a} + \sqrt{b} + \sqrt{c}$, the calculations just become absurd. Any kind of help would be enormously appreciated!  
 [/INST] You certainly have the right idea. Following Gerry Myerson's suggestion above, we have
\begin{align}
x&=\sqrt{a}+\sqrt{b}+\sqrt{c}\\
x-\sqrt{a}&=\sqrt{b}+\sqrt{c}\\
(x-\sqrt{a})^2=x^2-2x\sqrt{a}+a&=b+2\sqrt{bc}+c=(\sqrt{b}+\sqrt{c})^2\\
x^2+a-b-c&=2x\sqrt{a}+2\sqrt{bc}\\
\left(x^2+a-b-c\right)^2&=4x^2a+8x\sqrt{abc}+4bc\\
\left(x^2-a+b+c\right)^2-4x^2a-4bc&=8x\sqrt{abc}\\
\left(\left(x^2-a+b+c\right)^2-4x^2a-4bc\right)^2&=64abc\cdot x^2.
\end{align}
Expanding this will yield a polynomial with integer coefficients (assuming $a$, $b$, and $c$ are integers) which has a zero at $x=\sqrt{a}+\sqrt{b}+\sqrt{c}$.
</s>