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Beliebt Algebra >

erweitern (10x+2)^3(x-i)(8x^4-4)(9x^2+12x+3)

Lösung

erweitern

(10 x + 2)^{3} (x - i) (8 x^{4} - 4) (9 x^{2} + 12 x + 3)

Lösung

(72000 x^{10} + 139200 x^{9} + 90240 x^{8} - 32352 x^{6} - 69408 x^{5} - 45120 x^{4} + 26496 x^{7} - 13248 x^{3} - 1824 x^{2} - 96 x) + (96 - 72000 x^{9} - 139200 x^{8} - 90240 x^{7} + 32352 x^{5} + 69408 x^{4} + 45120 x^{3} - 26496 x^{6} + 13248 x^{2} + 1824 x) i

Schritte zur Lösung

(10 x + 2)^{3} (x - i) (8 x^{4} - 4) (9 x^{2} + 12 x + 3)

Schreibe

(10 x + 2)^{3}

um:

1000 x^{3} + 600 x^{2} + 120 x + 8

= (1000 x^{3} + 600 x^{2} + 120 x + 8) (x - i) (8 x^{4} - 4) (9 x^{2} + 12 x + 3)

Apply FOIL method

(x - i) (8 x^{4} - 4) : 8 x^{5} - 4 x - 8 x^{4} i + 4 i

= (1000 x^{3} + 600 x^{2} + 120 x + 8) (8 x^{5} - 4 x - 8 x^{4} i + 4 i) (9 x^{2} + 12 x + 3)

Schreibe

(1000 x^{3} + 600 x^{2} + 120 x + 8) (8 x^{5} - 4 x - 8 x^{4} i + 4 i)

um:

8000 x^{8} - 4000 x^{4} - 8000 x^{7} i + 4000 x^{3} i + 4800 x^{7} - 2400 x^{3} - 4800 x^{6} i + 2400 x^{2} i + 960 x^{6} - 480 x^{2} - 960 x^{5} i + 480 x i + 64 x^{5} - 32 x - 64 x^{4} i + 32 i

= (8000 x^{8} - 4000 x^{4} - 8000 x^{7} i + 4000 x^{3} i + 4800 x^{7} - 2400 x^{3} - 4800 x^{6} i + 2400 x^{2} i + 960 x^{6} - 480 x^{2} - 960 x^{5} i + 480 x i + 64 x^{5} - 32 x - 64 x^{4} i + 32 i) (9 x^{2} + 12 x + 3)

Setze Klammern

= 8000 x^{8} \cdot 9 x^{2} + 8000 x^{8} \cdot 12 x + 8000 x^{8} \cdot 3 - 4000 x^{4} \cdot 9 x^{2} - 4000 x^{4} \cdot 12 x - 4000 x^{4} \cdot 3 - 8000 x^{7} i 9 x^{2} - 8000 x^{7} i 12 x - 8000 x^{7} i 3 + 4000 x^{3} i 9 x^{2} + 4000 x^{3} i 12 x + 4000 x^{3} i 3 + 4800 x^{7} \cdot 9 x^{2} + 4800 x^{7} \cdot 12 x + 4800 x^{7} \cdot 3 - 2400 x^{3} \cdot 9 x^{2} - 2400 x^{3} \cdot 12 x - 2400 x^{3} \cdot 3 - 4800 x^{6} i 9 x^{2} - 4800 x^{6} i 12 x - 4800 x^{6} i 3 + 2400 x^{2} i 9 x^{2} + 2400 x^{2} i 12 x + 2400 x^{2} i 3 + 960 x^{6} \cdot 9 x^{2} + 960 x^{6} \cdot 12 x + 960 x^{6} \cdot 3 - 480 x^{2} \cdot 9 x^{2} - 480 x^{2} \cdot 12 x - 480 x^{2} \cdot 3 - 960 x^{5} i 9 x^{2} - 960 x^{5} i 12 x - 960 x^{5} i 3 + 480 x i 9 x^{2} + 480 x i 12 x + 480 x i 3 + 64 x^{5} \cdot 9 x^{2} + 64 x^{5} \cdot 12 x + 64 x^{5} \cdot 3 - 32 x \cdot 9 x^{2} - 32 x \cdot 12 x - 32 x \cdot 3 - 64 x^{4} i 9 x^{2} - 64 x^{4} i 12 x - 64 x^{4} i 3 + 32 i 9 x^{2} + 32 i 12 x + 32 i 3

Vereinfache

8000 x^{8} \cdot 9 x^{2} + 8000 x^{8} \cdot 12 x + 8000 x^{8} \cdot 3 - 4000 x^{4} \cdot 9 x^{2} - 4000 x^{4} \cdot 12 x - 4000 x^{4} \cdot 3 - 8000 x^{7} i 9 x^{2} - 8000 x^{7} i 12 x - 8000 x^{7} i 3 + 4000 x^{3} i 9 x^{2} + 4000 x^{3} i 12 x + 4000 x^{3} i 3 + 4800 x^{7} \cdot 9 x^{2} + 4800 x^{7} \cdot 12 x + 4800 x^{7} \cdot 3 - 2400 x^{3} \cdot 9 x^{2} - 2400 x^{3} \cdot 12 x - 2400 x^{3} \cdot 3 - 4800 x^{6} i 9 x^{2} - 4800 x^{6} i 12 x - 4800 x^{6} i 3 + 2400 x^{2} i 9 x^{2} + 2400 x^{2} i 12 x + 2400 x^{2} i 3 + 960 x^{6} \cdot 9 x^{2} + 960 x^{6} \cdot 12 x + 960 x^{6} \cdot 3 - 480 x^{2} \cdot 9 x^{2} - 480 x^{2} \cdot 12 x - 480 x^{2} \cdot 3 - 960 x^{5} i 9 x^{2} - 960 x^{5} i 12 x - 960 x^{5} i 3 + 480 x i 9 x^{2} + 480 x i 12 x + 480 x i 3 + 64 x^{5} \cdot 9 x^{2} + 64 x^{5} \cdot 12 x + 64 x^{5} \cdot 3 - 32 x \cdot 9 x^{2} - 32 x \cdot 12 x - 32 x \cdot 3 - 64 x^{4} i 9 x^{2} - 64 x^{4} i 12 x - 64 x^{4} i 3 + 32 i 9 x^{2} + 32 i 12 x + 32 i 3 : 96 i + 72000 x^{10} + 139200 x^{9} + 90240 x^{8} - 32352 x^{6} - 69408 x^{5} - 45120 x^{4} - 72000 x^{9} i - 139200 x^{8} i - 90240 x^{7} i + 26496 x^{7} + 32352 x^{5} i + 69408 x^{4} i + 45120 x^{3} i - 13248 x^{3} - 1824 x^{2} - 26496 x^{6} i + 13248 x^{2} i + 1824 x i - 96 x

= 96 i + 72000 x^{10} + 139200 x^{9} + 90240 x^{8} - 32352 x^{6} - 69408 x^{5} - 45120 x^{4} - 72000 x^{9} i - 139200 x^{8} i - 90240 x^{7} i + 26496 x^{7} + 32352 x^{5} i + 69408 x^{4} i + 45120 x^{3} i - 13248 x^{3} - 1824 x^{2} - 26496 x^{6} i + 13248 x^{2} i + 1824 x i - 96 x

Rewrite in standard complex form:

(72000 x^{10} + 139200 x^{9} + 90240 x^{8} - 32352 x^{6} - 69408 x^{5} - 45120 x^{4} + 26496 x^{7} - 13248 x^{3} - 1824 x^{2} - 96 x) + (96 - 72000 x^{9} - 139200 x^{8} - 90240 x^{7} + 32352 x^{5} + 69408 x^{4} + 45120 x^{3} - 26496 x^{6} + 13248 x^{2} + 1824 x) i

= (72000 x^{10} + 139200 x^{9} + 90240 x^{8} - 32352 x^{6} - 69408 x^{5} - 45120 x^{4} + 26496 x^{7} - 13248 x^{3} - 1824 x^{2} - 96 x) + (96 - 72000 x^{9} - 139200 x^{8} - 90240 x^{7} + 32352 x^{5} + 69408 x^{4} + 45120 x^{3} - 26496 x^{6} + 13248 x^{2} + 1824 x) i

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