- #1
ballzac
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- 0
The rest of this sheet of problems was a piece of cake, and I think this is meant to be one of the easier problems on it, but I'm not quite sure how to do it.
Find a fourth order polynomial with integer coefficients for which [tex]1+\sqrt{5}-2\sqrt{3}[/tex]
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 realized 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.
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 realized 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.