ข้อ 4 คล้ายเพชรยอดมงกุฎปีไหนจำไม่ได้ คำนวณตรงไปตรงมา
$ x=(\sqrt{2}-1)$
$ x^2=(\sqrt{2}-1)^2 = 2-2\sqrt{2}+1=3-2\sqrt{2} $
$ x^3=xx^2=(\sqrt{2}-1)(3-2\sqrt{2})=3\sqrt{2}-4-3+2\sqrt{2}=-7+5\sqrt{2} $
$ x^6=x^3x^3=(-7+5\sqrt{2})^2=49+50-70\sqrt{2}=99-70\sqrt{2}$
$ x^7=xx^6=(\sqrt{2}-1)(99-70\sqrt{2})=99\sqrt{2}-140-99+70\sqrt{2}=-239+169\sqrt{2} $
$ x^8=xx^7=(\sqrt{2}-1)(-239+169\sqrt{2})=-239\sqrt{2}+338+239-169\sqrt{2}=577-408\sqrt{2}$
$ x^9=xx^8=(\sqrt{2}-1)(577-408\sqrt{2})=577\sqrt{2}-816-577+408\sqrt{2}=-1393+985\sqrt{2}$
$ x^{10}=xx^9=(\sqrt{2}-1)(-1393+985\sqrt{2})=-1393\sqrt{2}+1970+1393-985\sqrt{2}=3363-2378\sqrt{2}$
ดังนั้น $1+6x+3x^2+x^6-2x^7-2x^7-2x^8+2x^9+x^{10}=4-1773\sqrt{2}$
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