5.2-bo'lim : ko'phadlarning nollari/ildizlari
5. \(x = r\) quyidagi ko‘phadning ildizidir. Qolgan ikkita ildizni toping va ko'phadni to'liq faktorlar shaklida yozing.
\[P\left( x \right) = {x^3} - 7{x^2} - 6x + 72 \mbox{ ; } r = 4\]Yechim
Biz \(x = 4\) ko'phadning ildizi ekanligini bilamiz va shuning uchun biz ko‘phadni quyidagicha yozishimiz mumkinligini bilamiz:
\[P\left( x \right) = \left( {x - 4} \right)Q\left( x \right)\]2-qadam
\(Q\left( x \right)\) ni topish uchun biz qilishimiz kerak bo'lgan narsa - tezkor sintetik bo'linish.
\[\begin{array}{*{20}{r}}{\left. {\underline {\,4 \,}}\! \right| }\\{}\\{}\end{array}\,\,\,\,\begin{array}{*{20}{r}}1&{ - 7}&{ - 6}&{72}\\{}&4&{ - 12}&{ - 72}\\\hline1&{ - 3}&{ - 18}&0\end{array}\]Bundan shuni ko'ramizki,
\[Q\left( x \right) = {x^2} - 3x - 18\]3-qadam
Endi biz \(P\left( x \right)\)-ni yozishimiz mumkin va bu \(Q\left( x \right)\) faktorini aniqlash uchun yetarlicha oddiy.
\[P\left( x \right) = \left( {x - 4} \right)\left( {{x^2} - 3x - 18} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\left( {x - 4} \right)\left( {x - 6} \right)\left( {x + 3} \right)}}\]4-qadam
Nihoyat, oldingi bosqichda \(P\left( x \right)\) faktorlanganidan biz ildiz/nollarning to'liq ro'yxatini ko'rishimiz mumkin:
\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = 4\,\,\,\,\,\,\,\,\,\,\,\,\,x = 6\,\,\,\,\,\,\,\,\,\,\,\,\,x = - 3}}\]