erweitern (1+x+x^2+x^3)^5

Lösung

erweitern

(1 + x + x^{2} + x^{3})^{5}

x^{15} + 5 x^{14} + 15 x^{13} + 35 x^{12} + 65 x^{11} + 101 x^{10} + 135 x^{9} + 155 x^{8} + 155 x^{7} + 135 x^{6} + 101 x^{5} + 65 x^{4} + 35 x^{3} + 15 x^{2} + 5 x + 1

Schritte zur Lösung

(1 + x + x^{2} + x^{3})^{5}

(1 + x + x^{2} + x^{3})^{5} = (1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3})

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

Multipliziere aus

(1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3}) (1 + x + x^{2} + x^{3}) : x^{15} + 5 x^{14} + 15 x^{13} + 35 x^{12} + 65 x^{11} + 101 x^{10} + 135 x^{9} + 155 x^{8} + 155 x^{7} + 135 x^{6} + 101 x^{5} + 65 x^{4} + 35 x^{3} + 15 x^{2} + 5 x + 1

= x^{15} + 5 x^{14} + 15 x^{13} + 35 x^{12} + 65 x^{11} + 101 x^{10} + 135 x^{9} + 155 x^{8} + 155 x^{7} + 135 x^{6} + 101 x^{5} + 65 x^{4} + 35 x^{3} + 15 x^{2} + 5 x + 1

e x p an d - 5 sin (3) (x + \frac{π}{3})

e x p an d \frac{3 ( 1 - 3 )}{2}

e x p an d \frac{( x - 1 ) ^{2}}{x + 2}

e x p an d 5 (x + Δ x)^{3}

e x p an d \frac{( 2 x + 1 )}{x ( x + 1 ) ( x ^{2} + 4 x + 6 )}