5.2-bo'lim : ko'phadlarning nollari/ildizlari
6. \(x = r\) quyidagi ko‘phadning ildizidir. Qolgan ikkita ildizni toping va ko'phadni to'liq faktorlar shaklida yozing.
\[P\left( x \right) = 3{x^3} + 16{x^2} - 33x + 14 \mbox{ ; } r = - 7\]Yechim
Biz \(x = - 7\) ko'phadning ildizi ekanligini bilamiz va shuning uchun biz ko‘phadni quyidagicha yozishimiz mumkinligini bilamiz:
\[P\left( x \right) = \left( {x + 7} \right)Q\left( x \right)\]2-qadam
\(Q\left( x \right)\) ni topish uchun biz qilishimiz kerak bo'lgan narsa - tezkor sintetik bo'linish.
\[\begin{array}{*{20}{r}}{\left. {\underline {\,{ - 7} \,}}\! \right| }\\{}\\{}\end{array}\,\,\,\,\begin{array}{*{20}{r}}3&{16}&{ - 33}&{14}\\{}&{ - 21}&{35}&{ - 14}\\\hline3&{ - 5}&2&0\end{array}\]Bundan shuni ko'ramizki,
\[Q\left( x \right) = 3{x^2} - 5x + 2\]3-qadam
Endi biz \(P\left( x \right)\)-ni yozishimiz mumkin va bu \(Q\left( x \right)\) faktorini aniqlash uchun yetarlicha oddiy.
\[P\left( x \right) = \left( {x + 7} \right)\left( {3{x^2} - 5x + 2} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\left( {x + 7} \right)\left( {3x - 2} \right)\left( {x - 1} \right)}}\]4-qadam
Nihoyat, oldingi bosqichda \(P\left( x \right)\) faktorlanganidan biz ildiz/nollarning to'liq ro'yxatini ko'rishimiz mumkin:
\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = - 7\,\,\,\,\,\,\,\,\,\,\,\,\,x = \frac{2}{3}\,\,\,\,\,\,\,\,\,\,\,\,\,x = 1}}\]