Îáðàçöû ðåøåíèé îñíîâíûõ çàäà÷
§ 1. Ìíîãî÷ëåíû íàä
îáëàñòüþ öåëîñòíîñòè
Çàäà÷à
2. Íàéäèòå êâàäðàò è êóá ìíîãî÷ëåíà f(x)=2x4 – 3x2 + x – 1.
Ðåøåíèå.
- 1. Èñïîëüçóÿ ôîðìóëû êâàäðàòàñóììû è êâàäðàòà ðàçíîñòè äâóõ
- âûðàæåíèé, âû÷èñëèì êâàäðàò ìíîãî÷ëåíà f(x).
(f(x) )2=(2x4–3x2+x–1)2=((2x4–3x2)+(x–1))2=(2x4–3x2)2+
+2×(2x4–3x2)×(x–1)+(x–1)2=(2x4)2–2×(2x4)×(3x2)+(3x2)2–
–2×(2x5–3õ3–2õ4+3õ2)+õ2–2×õ×1+12=4õ8–12õ6+9õ4–
– 4õ5 + 6õ3 + 4õ4 – 6õ2 + õ2 – 2õ + 1 = 4õ8 – 12õ6 – 4õ5 + 13õ4 + 6õ3 –
– 5õ2 –
2õ + 1.
- 2. Èñïîëüçóÿ ôîðìóëû êóáà ñóììû,êóáà ðàçíîñòè è êâàäðàòà
- ðàçíîñòè äâóõ âûðàæåíèé, âû÷èñëèì êóá ìíîãî÷ëåíà f(x).
(f(x) )3 = (2x4 – 3x2 + x – 1)3 = ((2x4 – 3x2) + (x – 1))3 = (2x4 – 3x2)3 +
+ 3 × (2x4 – 3x2)2 × (x – 1) + 3×(2x4 – 3õ2) × (x – 1)2 +(õ – 1)3 = (2x4)3 –
– 3×(2x4)2 × (3õ2) + 3×(2õ4)×(3õ2)2 – (3õ2)3 + (4õ8 – 12õ6 + 9õ4)×(3õ – 3) +
+ (6õ4 – 9õ2) × (õ2 – 2õ + 1) + õ3 – 3õ2 + 3õ – 1 = 8õ12 – 36õ10 + 54õ8 –
– 27õ6 + 12õ9 – 36õ7 + 27õ5 – 12õ8 + 36õ6 – 27õ4 + 6õ6 – 9õ4 – 12õ5+
+ 18õ3 + 6õ4 – 9õ2 + õ3 – 3õ2 + 3õ – 1 = 8õ12 – 36õ10 + 12õ9 + 42õ8 –
– 36õ7 + 15õ6
+ 15õ5 – 30õ4 + 19õ3 – 12õ2 + 3õ –
1.
Îòâåò:
(f(x) )2 = 4õ8 – 12õ6 –
4õ5 + 13õ4 + 6õ3 – 5õ2 – 2õ + 1;
(f(x) )3 = 8õ12 – 36õ10 + 12õ9 + 42õ8 – 36õ7 + 15õ6 + 15õ5 –
– 30õ4 + 19õ3 – 12õ2
+ 3õ – 1.