Bu ko'phad uchun barcha mumkin bo'lgan ratsional nollarning ro'yxati.

\[\begin{align*}\frac{{ \pm 1}}{{ \pm 1}} & = \pm 1 & \hspace{0.25in}\frac{{ \pm 2}}{{ \pm 1}} & = \pm 2 & \hspace{0.25in}\frac{{ \pm 3}}{{ \pm 1}} & = \pm 3 & \hspace{0.25in}\frac{{ \pm 6}}{{ \pm 1}} & = \pm 6 & \hspace{0.25in}\frac{{ \pm 9}}{{ \pm 1}} & = \pm 9 & \hspace{0.25in}\frac{{ \pm 18}}{{ \pm 1}} & = \pm 18\\ & \\ \frac{{ \pm 1}}{{ \pm 2}} & = \frac{{ \pm 1}}{{ \pm 2}} & \hspace{0.25in}\frac{{ \pm 2}}{{ \pm 2}} & = \pm 1 & \hspace{0.25in}\frac{{ \pm 3}}{{ \pm 2}} & = \frac{{ \pm 3}}{{ \pm 2}} & \hspace{0.25in}\frac{{ \pm 6}}{{ \pm 2}} & = \pm 3 & \hspace{0.25in}\frac{{ \pm 9}}{{ \pm 2}} & = \frac{{ \pm 9}}{{ \pm 2}} & \hspace{0.25in}\frac{{ \pm 18}}{{ \pm 2}} & = \pm 9\end{align*}\]

Demak, ko'phad uchun jami 18 ta nol mavjud.

Endi biz sintetik bo'linish ishini boshlashimiz kerak. Biz birinchi navbatda "kichik" butun sonlardan boshlaymiz.

\[\,\,\begin{array}{r|rrrl} {} & 2 & -13 & 3 & 18 \\ \hline -1 & 2 & -15 & 18 & 0=f\left( -1 \right)=0!! \\ \end{array}\]

Endi biz \(x = - 1\) nol ekanligini bilamiz va ko'phadni quyidagicha yozishimiz mumkin:

\[f\left( x \right) = 2{x^3} - 13{x^2} + 3x + 18 = \left( {x + 1} \right)\left( {2{x^2} - 15x + 18} \right)\]

Biz bu jarayonni davom ettirishimiz mumkin, ammo bizda ikkinchi omil uchun kvadratik bor va biz buni shunchaki faktorlashimiz mumkin, shuning uchun ko'phadning to'liq faktorlangan shakli,

\[f\left( x \right) = 2{x^3} - 13{x^2} + 3x + 18 = \left( {x + 1} \right)\left( {2x - 3} \right)\left( {x - 6} \right)\]

To'liq faktorlangan shakldan biz asl ko'phad uchun quyidagi nollar to'plamini olamiz.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = - 1\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}x = \frac{3}{2}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}x = 6}}\]