展開する ((3x+1)^4(x-4))/((x^2+1)^3sin(x))

解

展開する

\frac{( 3 x + 1 ) ^{4} ( x - 4 )}{( x ^{2} + 1 ) ^{3} sin ( x )}

\frac{81 x ^{5} - 216 x ^{4} - 378 x ^{3} - 204 x ^{2} - 47 x - 4}{x ^{6} sin ( x ) + 3 x ^{4} sin ( x ) + 3 x ^{2} sin ( x ) + sin ( x )}

解答ステップ

\frac{( 3 x + 1 ) ^{4} ( x - 4 )}{( x ^{2} + 1 ) ^{3} sin ( x )}

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

= \frac{( 3 x + 1 ) ^{4} ( x - 4 )}{sin ( x ) ( x ^{6} + 3 x ^{4} + 3 x ^{2} + 1 )}

(3 x + 1)^{4} = 81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1

= \frac{( 81 x ^{4} + 108 x ^{3} + 54 x ^{2} + 12 x + 1 ) ( x - 4 )}{sin ( x ) ( x ^{6} + 3 x ^{4} + 3 x ^{2} + 1 )}

拡張

(x^{6} + 3 x^{4} + 3 x^{2} + 1) sin (x) : x^{6} sin (x) + 3 x^{4} sin (x) + 3 x^{2} sin (x) + sin (x)

= \frac{( 81 x ^{4} + 108 x ^{3} + 54 x ^{2} + 12 x + 1 ) ( x - 4 )}{x ^{6} sin ( x ) + 3 x ^{4} sin ( x ) + 3 x ^{2} sin ( x ) + sin ( x )}

拡張

(81 x^{4} + 108 x^{3} + 54 x^{2} + 12 x + 1) (x - 4) : 81 x^{5} - 216 x^{4} - 378 x^{3} - 204 x^{2} - 47 x - 4

= \frac{81 x ^{5} - 216 x ^{4} - 378 x ^{3} - 204 x ^{2} - 47 x - 4}{x ^{6} sin ( x ) + 3 x ^{4} sin ( x ) + 3 x ^{2} sin ( x ) + sin ( x )}