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Number Theory: Interesting Results

One root of ...

Question

One root of $a x^{4} + b x^{3} + c x^{2} + d x + e = 0$ , where $a, b, c, d, e$ are rational numbers, is $\sqrt{2} + \sqrt{3}$ , then the other real roots are:

A

\sqrt{2} - \sqrt{3}, 2, 5

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B

\sqrt{2} - \sqrt{3}, - 2, 5

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C

\sqrt{2} - \sqrt{3}, - \sqrt{2} + \sqrt{3}, 5

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D

\sqrt{2} - \sqrt{3}, - \sqrt{2} + \sqrt{3}, - \sqrt{2} - \sqrt{3}

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Solution

The correct option is C $\sqrt{2} - \sqrt{3}, - \sqrt{2} + \sqrt{3}, - \sqrt{2} - \sqrt{3}$
$C o e f f i c i e n t s a r e r a t i o n a l, h e n c e a l l t h e i r r a t i o n a l i t i e s m u s t b e c a n c e l l e d H e n c e t h e r o o t s a r e \sqrt{2} + \sqrt{3}, \sqrt{2} - \sqrt{3}, - \sqrt{2} + \sqrt{3}, - \sqrt{2} - \sqrt{3}$

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