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!  

Subtract $\sqrt a$ from both sides, square both sides: now you have $\sqrt a$ on one side, $\sqrt{bc}$ on the other. Solve for $\sqrt a$, square both sides: now you have only $\sqrt{bc}$. Solve for $\sqrt{bc}$, square both sides, voila! all square roots gone. 
If you need to know the other zeros, the full set is $\pm\sqrt a\pm\sqrt b\pm\sqrt c$ where the signs are to be taken independently of each other (making 8 zeros in all). 
