5.2-bo'lim : ko'phadlarning nollari/ildizlari
4. \(x = r\) quyidagi ko‘phadning ildizidir. Qolgan ikkita ildizni toping va ko'phadni to'liq faktorlar shaklida yozing.
\[P\left( x \right) = {x^3} - 6{x^2} - 16x \mbox{ ; }r = - 2\]Yechim
Biz \(x = - 2\) ko'phadning ildizi ekanligini bilamiz va shuning uchun biz ko‘phadni quyidagicha yozishimiz mumkinligini bilamiz:
\[P\left( x \right) = \left( {x + 2} \right)Q\left( x \right)\]2-qadam
\(Q\left( x \right)\) ni topish uchun biz qilishimiz kerak bo'lgan narsa - tezkor sintetik bo'linish.
\[\begin{array}{*{20}{r}}{\left. {\underline {\,{ - 2} \,}}\! \right| }\\{}\\{}\end{array}\,\,\,\,\begin{array}{*{20}{r}}1&{ - 6}&{ - 16}&0\\{}&{ - 2}&{16}&0\\\hline1&{ - 8}&0&0\end{array}\]Bundan shuni ko'ramizki,
\[Q\left( x \right) = {x^2} - 8x\]3-qadam
Endi biz \(P\left( x \right)\)-ni yozishimiz mumkin va bu \(Q\left( x \right)\) faktorini aniqlash uchun yetarlicha oddiy.
\[P\left( x \right) = \left( {x + 2} \right)\left( {{x^2} - 8x} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{x\left( {x + 2} \right)\left( {x - 8} \right)}}\]4-qadam
Nihoyat, oldingi bosqichda \(P\left( x \right)\) faktorlanganidan biz ildiz/nollarning to'liq ro'yxatini ko'rishimiz mumkin:
\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = 0\,\,\,\,\,\,\,\,\,\,\,\,\,x = - 2\,\,\,\,\,\,\,\,\,\,\,\,\,x = 8}}\]