erweitern (x^4+x^3+x^2+x+1)^4

Lösung

erweitern

(x^{4} + x^{3} + x^{2} + x + 1)^{4}

x^{16} + 4 x^{15} + 10 x^{14} + 20 x^{13} + 35 x^{12} + 52 x^{11} + 68 x^{10} + 80 x^{9} + 85 x^{8} + 80 x^{7} + 68 x^{6} + 52 x^{5} + 35 x^{4} + 20 x^{3} + 10 x^{2} + 4 x + 1

Schritte zur Lösung

(x^{4} + x^{3} + x^{2} + x + 1)^{4}

Vereinfache

(x^{4} + x^{3} + x^{2} + x + 1)^{4} : (x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1)

= (x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1)

Schreibe

(x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1)

um:

x^{8} + 2 x^{7} + 3 x^{6} + 4 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1

= (x^{8} + 2 x^{7} + 3 x^{6} + 4 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1) (x^{4} + x^{3} + x^{2} + x + 1) (x^{4} + x^{3} + x^{2} + x + 1)

Schreibe

(x^{8} + 2 x^{7} + 3 x^{6} + 4 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1) (x^{4} + x^{3} + x^{2} + x + 1)

um:

x^{12} + 3 x^{11} + 6 x^{10} + 10 x^{9} + 15 x^{8} + 18 x^{7} + 19 x^{6} + 18 x^{5} + 15 x^{4} + 10 x^{3} + 6 x^{2} + 3 x + 1

= (x^{12} + 3 x^{11} + 6 x^{10} + 10 x^{9} + 15 x^{8} + 18 x^{7} + 19 x^{6} + 18 x^{5} + 15 x^{4} + 10 x^{3} + 6 x^{2} + 3 x + 1) (x^{4} + x^{3} + x^{2} + x + 1)

Setze Klammern

= x^{12} x^{4} + x^{12} x^{3} + x^{12} x^{2} + x^{12} x + x^{12} \cdot 1 + 3 x^{11} x^{4} + 3 x^{11} x^{3} + 3 x^{11} x^{2} + 3 x^{11} x + 3 x^{11} \cdot 1 + 6 x^{10} x^{4} + 6 x^{10} x^{3} + 6 x^{10} x^{2} + 6 x^{10} x + 6 x^{10} \cdot 1 + 10 x^{9} x^{4} + 10 x^{9} x^{3} + 10 x^{9} x^{2} + 10 x^{9} x + 10 x^{9} \cdot 1 + 15 x^{8} x^{4} + 15 x^{8} x^{3} + 15 x^{8} x^{2} + 15 x^{8} x + 15 x^{8} \cdot 1 + 18 x^{7} x^{4} + 18 x^{7} x^{3} + 18 x^{7} x^{2} + 18 x^{7} x + 18 x^{7} \cdot 1 + 19 x^{6} x^{4} + 19 x^{6} x^{3} + 19 x^{6} x^{2} + 19 x^{6} x + 19 x^{6} \cdot 1 + 18 x^{5} x^{4} + 18 x^{5} x^{3} + 18 x^{5} x^{2} + 18 x^{5} x + 18 x^{5} \cdot 1 + 15 x^{4} x^{4} + 15 x^{4} x^{3} + 15 x^{4} x^{2} + 15 x^{4} x + 15 x^{4} \cdot 1 + 10 x^{3} x^{4} + 10 x^{3} x^{3} + 10 x^{3} x^{2} + 10 x^{3} x + 10 x^{3} \cdot 1 + 6 x^{2} x^{4} + 6 x^{2} x^{3} + 6 x^{2} x^{2} + 6 x^{2} x + 6 x^{2} \cdot 1 + 3 x x^{4} + 3 x x^{3} + 3 x x^{2} + 3 xx + 3 x \cdot 1 + 1 \cdot x^{4} + 1 \cdot x^{3} + 1 \cdot x^{2} + 1 \cdot x + 1 \cdot 1

Vereinfache

x^{12} x^{4} + x^{12} x^{3} + x^{12} x^{2} + x^{12} x + x^{12} \cdot 1 + 3 x^{11} x^{4} + 3 x^{11} x^{3} + 3 x^{11} x^{2} + 3 x^{11} x + 3 x^{11} \cdot 1 + 6 x^{10} x^{4} + 6 x^{10} x^{3} + 6 x^{10} x^{2} + 6 x^{10} x + 6 x^{10} \cdot 1 + 10 x^{9} x^{4} + 10 x^{9} x^{3} + 10 x^{9} x^{2} + 10 x^{9} x + 10 x^{9} \cdot 1 + 15 x^{8} x^{4} + 15 x^{8} x^{3} + 15 x^{8} x^{2} + 15 x^{8} x + 15 x^{8} \cdot 1 + 18 x^{7} x^{4} + 18 x^{7} x^{3} + 18 x^{7} x^{2} + 18 x^{7} x + 18 x^{7} \cdot 1 + 19 x^{6} x^{4} + 19 x^{6} x^{3} + 19 x^{6} x^{2} + 19 x^{6} x + 19 x^{6} \cdot 1 + 18 x^{5} x^{4} + 18 x^{5} x^{3} + 18 x^{5} x^{2} + 18 x^{5} x + 18 x^{5} \cdot 1 + 15 x^{4} x^{4} + 15 x^{4} x^{3} + 15 x^{4} x^{2} + 15 x^{4} x + 15 x^{4} \cdot 1 + 10 x^{3} x^{4} + 10 x^{3} x^{3} + 10 x^{3} x^{2} + 10 x^{3} x + 10 x^{3} \cdot 1 + 6 x^{2} x^{4} + 6 x^{2} x^{3} + 6 x^{2} x^{2} + 6 x^{2} x + 6 x^{2} \cdot 1 + 3 x x^{4} + 3 x x^{3} + 3 x x^{2} + 3 xx + 3 x \cdot 1 + 1 \cdot x^{4} + 1 \cdot x^{3} + 1 \cdot x^{2} + 1 \cdot x + 1 \cdot 1 : x^{16} + 4 x^{15} + 10 x^{14} + 20 x^{13} + 35 x^{12} + 52 x^{11} + 68 x^{10} + 80 x^{9} + 85 x^{8} + 80 x^{7} + 68 x^{6} + 52 x^{5} + 35 x^{4} + 20 x^{3} + 10 x^{2} + 4 x + 1

= x^{16} + 4 x^{15} + 10 x^{14} + 20 x^{13} + 35 x^{12} + 52 x^{11} + 68 x^{10} + 80 x^{9} + 85 x^{8} + 80 x^{7} + 68 x^{6} + 52 x^{5} + 35 x^{4} + 20 x^{3} + 10 x^{2} + 4 x + 1

s im pl i f y 5^{2} \cdot π

s im pl i f y f (g (- 3))

e x p an d (x - 1)^{3} (x + 1)

f a c t or 4 m^{3} n - 2 mn + 6 m

s im pl i f y (4 m^{- 4}) (2 n^{5})