<prompt> Bidhaa ya polynomial ya fomu1 - zb_1 1 - z2b_2 1 - z3b_3 1 - z4b_4 1 - z5 b_5 dotsm 1 - z32b_32,ambapo b_k ni nambari chanya, ina sifa ya kushangaza kwamba ikiwa tutaizidisha na kutupa masharti yote yanayohusisha. z kwa nguvu kubwa kuliko 32, kilichosalia ni 1 - 2z. Determine b_32.Unaweza kuingiza jibu lako kwa kutumia nukuu ya kielelezo.  <response>: Acha gz 1 - zb_1 1 - z2b_2 1 - z3b_3 1 - z4b_4 1 - z5b_5 dotsm 1 - z32b_32. Kwa kuwa gz inapungua hadi 1 - 2z ikiwa tutaondoa mamlaka yote ya z ambazo ni z33 au zaidi, tunaandikagz sawa na 1 - 2z pmodz33.Kishaanzaaligng-z 1 zb_1 1 - z2b_2 1 z3b_3 1 - z4b_4 1 z5b_5 dotsm 1 - z32b_32 sawa 1 2z pmodz33,mwishoalignhivyoanzaaligngz g-z 1 - z2b_1 2b_2 1 - z42b_4 1 - z6b_3 2b_6 1 - z82b_8 dotsm 1 - z30b_15 2b_30 1 - z322b_32 sawa 1 2z1 - 2z sawa 1 - 22 z2 pmodz33.endalignHebu g_1z2 gz g-z, hivyoanzaaligng_1z 1 - zc_1 1 - z2c_2 1 - z3c_3 1 - z4c_4 dotsm 1 - z16c_16 sawa na 1 - 22 z pmodz17,endalignambapo c_i b_i 2b_2i ikiwa i ni isiyo ya kawaida, na c_i 2b_2i ikiwa i ni sawa. Hasa, c_16 2b_32.Kishaanzaaligng_1z g_1-z 1 - z2c_1 2c_2 1 - z42c_4 1 - z6c_3 2c_6 1 - z82c_8 dotsm 1 - z14c_7 2c_14 1 - z162c_16 equiv 1 - 22 z1 22 z sawa na 1 - 24 z2 pmodz17.endalignKwa hivyo, acha g_2z2 g_1z g_1-z, hivyoanzaaligng_2 z 1 - zd_1 1 - z2d_2 1 - z3d_3 1 - zd_4 dotsm 1 - z7d_7 1 - z8d_8 sawa na 1 - 24 z pmodz9,endalignambapo d_i c_i 2c_2i ikiwa i ni isiyo ya kawaida, na d_i 2c_2i ikiwa i ni sawa. Hasa, d_8 2c_16.Vile vile, tunapata polynomial g_3z kama hiyog_3z 1 - ze_1 1 - z2e_2 1 - z3e_3 1 - ze_4 sawa na 1 - 28 z pmodz5,na polynomial g_4z kiasi kwambag_4z 1 - zf_1 1 - z2f_2 sawa na 1 - 216 z pmodz3.Kupanua, sisi pataanzaaligng_4z 1 - zf_1 1 - z2f_2 kushoto 1 - f_1 z binomf_12 z2 - dotsb kulia kushoto 1 - f_2 z2 dotsb kulia 1 - f_1 z kushoto binomf_12 - f_2 kulia z2 dotsb.endalignKwa hivyo, f_1 216 na binomf_12 - f_2 0, hivyof_2 binomf_12 binom2162 frac216 216 - 12 2 31 - 215.Tuna hiyo f_2 2e_4 4d_8 8c_16 16b_32, hivyob_32 fracf_216 227 - 211.Tunamwachia msomaji kutafuta polynomia ambayo inakidhi sharti lililotolewa. 