<prompt> Pata polynomial px, na coefficients halisi, kama hiyopx3 - px3 - 2 px2 12kwa nambari zote halisi x.  <response>: Hebupx a_n xn a_n - 1 xn - 1 vitone a_1 x a_0,ambapo a_n neq 0. Kishaanzaalignpx3 - px3 - 2 a_n x3n a_n - 1 x3n - 3 dotsb - a_n x3 - 2 n - a_n - 1 x3 - 2n - 1 - dotsb a_n x3n a_n - 1 x3n - 3 dotsb - a_n x3n - 2na_n x3n - 3 - dotsb - a_n - 1 x3n - 3 - dotsb 2n a_n x3n - 3 dotsb.endalignKwa hivyo, kiwango cha px3 - px3 - 2 ni 3n - 3.Kiwango cha px2 12 ni 2n, hivyo 3n - 3 2n, ambayo ina maana n 3.Hebu px ax3 bx2 cx d. Kishaanzaalignpx3 - px3 - 2 ax9 bx6 cx3 d - ax3 - 23 bx3 - 2 2 cx3 - 2 d 6ax6 -12a 4b x3 8a - 4b 2c,mwishoalignnapx2 12 a2 x6 2abx5 2ac b2 x4 2ad 2bc x3 2bd c2 x2 2cdx d2 12.Tukilinganisha vigawo, tunapataanzaaligna2 6a, 2ab 0, 2ac b2 0, 2ad 2bc -12a 4b, 2bd c2 0, 2cd 0, d2 12 8a - 4b 2c.endalignKutoka kwa mlinganyo a2 6a, a 0 au a 6. Lakini kwa kuwa a ni mgawo unaoongoza, a haiwezi kuwa 0, kwa hivyo a 6.Kutoka kwa mlinganyo 2ab 0, b 0.Kisha mlinganyo 2ac b2 0 inakuwa 12c 0, hivyo c 0.Kisha equation 2ad 2bc -12a 4b inakuwa 12d -72, so d -6. Kumbuka kwamba a,b,c,d 6,0,0,- 6 hutosheleza milinganyo yote.Kwa hiyo, px 6x3 - 6. 