<prompt> Polynomial Px ni monic, quartic polynomia na coefficients halisi, na mbili ya mizizi yake ni cos theta i sin theta na sin theta i cos theta, ambapo 0 theta fracpi4. Mizizi minne ya Px inapopangwa katika ndege changamano, huunda pembe nne ambayo eneo lake ni sawa na nusu ya P0 . Tafuta jumla ya mizizi minne.  <response>: Kwa kuwa Px ya polynomial ina coefficients halisi, ikiwa z ni mzizi usio halisi wa Px, basi hivyo ni muunganisho wake overlinez. Kwa hivyo, mizizi mingine miwili ya Px ni cos theta - i sin theta na sin theta - i cos theta. Tunapopanga mizizi minne yote iko kwenye duara la kitengo, sisi kupata trapezoid.asyukubwa wa kitengo 2 cm;jozi A, B, C, D;A dir30;B dir60;C dir-60;D dir-30;kujazaA--B--C--D--mzunguko,kijivu0.7;choraMduara0,0,1;chora-1.2,0--1.2,0;chora0,-1.2--0,1.2;nuktacos theta i sin theta, A, A;nuktasin theta i cos theta, B, B;nuktasin theta - i cos theta, C, C;nuktacos theta - i sin theta, D, D;asyEneo la trapezoid hii nianzaalignfrac2 cos theta 2 sin theta2 cdot cos theta - sin theta cos theta sin thetacos theta - dhambi theta cos2 theta - sin2 theta cos 2 theta.endalignMonic quartic Px nianzaalignx - cos theta i sin thetax - cos theta - i sin thetax - sin theta i cos theta x - dhambi theta - i cos theta x2 - 2x cos theta 1x2 - 2x sin theta 1.endalignKisha P0 1, kwa hivyo eneo la pembe nne ni frac12. Kwa hivyo,cos 2 theta frac12.Kwa kuwa 0 2 theta fracpi2, lazima tuwe na 2 theta fracpi 3, au theta fracpi6.Jumla ya mizizi minne basi ni 2 cos theta 2 sin theta 1 sqrt3. 