<prompt> Tafuta polynomial isiyobadilika Px kama hiyoPPx x2 x 1 Px.  <response>: Acha d iwe shahada ya Px. Kisha shahada ya PPx ni d2, na shahada ya x2 x 1 P x ni d 2, hivyod2 d 2.Kisha d2 - d - 2 d - 2d 1 0. Kwa kuwa d ni chanya, d 2.Hebu Px ax2 bx c. KishaanzaalignPPx ashoka2 bx c2 bshoka2 bx c c a3 x4 2a2 bx3 ab2 2a2 c ab x2 2abc b2 x ac2 bc cmwishoalignnax2 x 1shoka2 bx c shoka4 a b x3 a b c x2 b c x c.Tunapata kwa kulinganisha hesabuanzaaligna3 a, 2a2 b a b, ab2 2a2 c ab a b c, 2abc b2 b c, ac2 bc c c.endalignKutoka a3 a, a3 - a aa - 1a 1 0, hivyo a ni 0, 1, au -1 . Lakini a ndio mgawo unaoongoza, kwa hivyo a haiwezi kuwa 0, kumaanisha a ni 1 au -1.Ikiwa a 1, basi 2b 1 b, hivyo b 1. Kisha1 2c 1 1 1 c,hivyo c 0. Kumbuka kwamba a,b,c 1,1,0 inatosheleza milinganyo yote.Ikiwa a -1, basi 2b -1 b, hivyo b -1. Kisha-1 2c 1 -1 - 1 c,hivyo c -2. Lakini basi mlinganyo ac2 bc c c hauridhiki.Kwa hivyo, a,b,c 1,1,0, na Px x2 x. 