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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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