- #1

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## Homework Statement

Find a fourth order polynomial with integer coefficients for which [tex]1+\sqrt{5}-2\sqrt{3}[/tex]

## The Attempt at a Solution

I tried rearranging it thus

[tex] (x-1)^2=17-4\sqrt{15}[/tex]

wasn't sure what to do here, but fourth order is required, so I tried squaring both sides again...

[tex](x-1)^4=529-8\sqrt{15}[/tex]

Now I don't know how to get rid of the surd. I tried expanding out the left hand side, which didn't help. Hopefully it's not my arithmetic, as that would be a little embarrassing, but I have tried several times and not resolved it. Any help pointing me in the right direction would be greatly appreciated.

Hmm...I just realised that if [tex]\sqrt{15}[/tex] is rational then I can multiply the whole thing by the denominator. So I will try to figure that out if it is rational, however my intuition says it is not (my intuition has failed me in the past, so I will try to prove it), and it may be difficult to find what it is as a ratio if it is rational.