erweitern (x^3-4x^2+2x+5)/(x(x+3)^6(x-2)^6(x^2+9))

Lösung

erweitern

\frac{x ^{3} - 4 x ^{2} + 2 x + 5}{x ( x + 3 ) ^{6} ( x - 2 ) ^{6} ( x ^{2} + 9 )}

\frac{x ^{3} - 4 x ^{2} + 2 x + 5}{x ^{15} + 6 x ^{14} - 12 x ^{13} - 106 x ^{12} + 6 x ^{11} + 366 x ^{10} + 496 x ^{9} + 5418 x ^{8} - 4311 x ^{7} - 62964 x ^{6} + 35964 x ^{5} + 264384 x ^{4} - 198288 x ^{3} - 419904 x ^{2} + 419904 x}

Schritte zur Lösung

\frac{x ^{3} - 4 x ^{2} + 2 x + 5}{x ( x + 3 ) ^{6} ( x - 2 ) ^{6} ( x ^{2} + 9 )}

(x + 3)^{6} = x^{6} + 18 x^{5} + 135 x^{4} + 540 x^{3} + 1215 x^{2} + 1458 x + 729

= \frac{x ^{3} - 4 x ^{2} + 2 x + 5}{x ( x ^{6} + 18 x ^{5} + 135 x ^{4} + 540 x ^{3} + 1215 x ^{2} + 1458 x + 729 ) ( x - 2 ) ^{6} ( x ^{2} + 9 )}

(x - 2)^{6} = x^{6} - 12 x^{5} + 60 x^{4} - 160 x^{3} + 240 x^{2} - 192 x + 64

= \frac{x ^{3} - 4 x ^{2} + 2 x + 5}{x ( x ^{6} + 18 x ^{5} + 135 x ^{4} + 540 x ^{3} + 1215 x ^{2} + 1458 x + 729 ) ( x ^{6} - 12 x ^{5} + 60 x ^{4} - 160 x ^{3} + 240 x ^{2} - 192 x + 64 ) ( x ^{2} + 9 )}

Multipliziere aus

x (x^{6} + 18 x^{5} + 135 x^{4} + 540 x^{3} + 1215 x^{2} + 1458 x + 729) (x^{6} - 12 x^{5} + 60 x^{4} - 160 x^{3} + 240 x^{2} - 192 x + 64) (x^{2} + 9) : x^{15} + 6 x^{14} - 12 x^{13} - 106 x^{12} + 6 x^{11} + 366 x^{10} + 496 x^{9} + 5418 x^{8} - 4311 x^{7} - 62964 x^{6} + 35964 x^{5} + 264384 x^{4} - 198288 x^{3} - 419904 x^{2} + 419904 x

= \frac{x ^{3} - 4 x ^{2} + 2 x + 5}{x ^{15} + 6 x ^{14} - 12 x ^{13} - 106 x ^{12} + 6 x ^{11} + 366 x ^{10} + 496 x ^{9} + 5418 x ^{8} - 4311 x ^{7} - 62964 x ^{6} + 35964 x ^{5} + 264384 x ^{4} - 198288 x ^{3} - 419904 x ^{2} + 419904 x}

e x p an d (0, 3) y (0, - 3)

e x p an d \frac{7 x + 9}{( x + 2 ) ( 3 x + 1 )}

e x p an d e^{x} cos (x) (e^{x} sin (x) + cos (x) e^{x})

e x p an d (t^{2} - e^{3 t}) U (t - 2)

s im pl i f y 1 - \frac{1}{2} + \frac{3}{32} (4 (0.7) - \frac{0. 7 ^{3}}{3})