If $x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of $x^{4} + a x^{3} + b x^{2} + c x + d = 0$ where $a, b, c, d$ are integers, what is the value of $| a + b + c + d |$ ?

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Solution

$x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of equation:
$x^{4} + a x^{3} + b x^{2} + c x + d = 0$ ; where $a, b, c, d \in Z$
Now, $x - \sqrt{2} = \sqrt{3} + \sqrt{6}$
Squaring both sides, we get
$(x - \sqrt{2})^{2} = (\sqrt{3} + \sqrt{6})^{2}$
$\Rightarrow x^{2} - 2 \sqrt{2 x} + 2 = 9 + 6 \sqrt{2}$
$\Rightarrow x^{2} - 7 = 2 \sqrt{2} (3 + x)$
Squaring again both sides, we get
$\Rightarrow x^{4} - 14 x^{2} + 49 = 8 (x^{2} + 6 x + 9)$
$\Rightarrow x^{4} - 22 x^{2} - 48 x - 23 = 0$
$∴ a = 0, b = - 22, c = - 48$ and $d = - 23$
$∴ | a + b + c + d | = 93$

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