Îáðàçöû ðåøåíèé îñíîâíûõ çàäà÷

§ 3. Ìíîãî÷ëåíû íàä ÷èñëîâûìè ïîëÿìè

                Çàäà÷à 26. Íàéäèòå ïðèâåäåííûé ìíîãî÷ëåííàèìåíüøåé
ñòåïåíè ñ êîìïëåêñíûìè êîýôôèöèåíòàìè, èìåþùèé íàáîð êîðíåé:
ïðîñòûåêîðíè 1, (–2) è 3 è äâóõ êðàòíûé êîðåíü   (1 – 3i).
 
              Ðåøåíèå.
              Èç òåîðèè èçâåñòíî, ÷òî åñëè õ = ñ – êîðåíü ìíîãî÷ëåíà
f(x), òî f(x)  äåëèòñÿ íà äâó÷ëåí (õ – ñ).Ñëåäîâàòåëüíî, èñêîìûé ìíîãî÷ëåí
äîëæåí äåëèòüñÿ íà (õ – 1), íà (õ + 2), íà (õ– 3) è íà (õ– (1 – 3i))2.
               
              À òàê êàê îí äîëæåí èìåòüíàèìåíüøóþ ñòåïåíü, òî
f(x) = (õ – 1)×(õ + 2)×(õ – 3)×(õ –(1 – 3i))2 = (õ2 + õ – 2)×(õ – 3)×(õ2 – 2 ×
× (1 – 3i)õ + (1 – 3i)2) = (õ3 – 3õ2 + õ2 – 3õ – 2õ + 6)×(õ2 +(–2 + 6i)õ + 
+ 1 – 6i + 9i2) = (õ3 – 2õ2 – 5õ + 6)×(õ2 +(–2 + 6i)õ – 8 – 6i) = õ5
+ (–2 + 6i)õ4 – 8õ – 6iõ3 – 2õ4 +(4 – 12i)õ3 + 16õ2 + 12iõ2 – 5õ2
 + (10 – 30i)õ2 + 40õ + 30iõ + 6õ2 + (–12 + 36i)õ – 48 – 36i = õ5 + (–4 + 6i)õ4+
+ (–9 – 18i)õ3 + (32 – 18i)õ2 + (28 + 66i)õ – 48 – 36i.   
 
             Îòâåò:
f(x) = õ5 + (–4 + 6i)õ4 +(–9 – 18i)õ3 + (32 – 18i)õ2 + (28 + 66i)õ – 48 – 36i

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