展開する (((x^2+3)^5(x-1)^2))/(x^4(x^3+5)^3)

解

展開する

\frac{( ( x ^{2} + 3 ) ^{5} ( x - 1 ) ^{2} )}{x ^{4} ( x ^{3} + 5 ) ^{3}}

\frac{x ^{12} - 2 x ^{11} + 16 x ^{10} - 30 x ^{9} + 105 x ^{8} - 180 x ^{7} + 360 x ^{6} - 540 x ^{5} + 675 x ^{4} - 810 x ^{3} + 648 x ^{2} - 486 x + 243}{x ^{13} + 15 x ^{10} + 75 x ^{7} + 125 x ^{4}}

解答ステップ

\frac{( x ^{2} + 3 ) ^{5} ( x - 1 ) ^{2}}{x ^{4} ( x ^{3} + 5 ) ^{3}}

(x^{3} + 5)^{3} = x^{9} + 15 x^{6} + 75 x^{3} + 125

= \frac{( x ^{2} + 3 ) ^{5} ( x - 1 ) ^{2}}{x ^{4} ( x ^{9} + 15 x ^{6} + 75 x ^{3} + 125 )}

(x^{2} + 3)^{5} = x^{10} + 15 x^{8} + 90 x^{6} + 270 x^{4} + 405 x^{2} + 243

= \frac{( x ^{10} + 15 x ^{8} + 90 x ^{6} + 270 x ^{4} + 405 x ^{2} + 243 ) ( x - 1 ) ^{2}}{x ^{4} ( x ^{9} + 15 x ^{6} + 75 x ^{3} + 125 )}

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

= \frac{( x ^{10} + 15 x ^{8} + 90 x ^{6} + 270 x ^{4} + 405 x ^{2} + 243 ) ( x ^{2} - 2 x + 1 )}{x ^{4} ( x ^{9} + 15 x ^{6} + 75 x ^{3} + 125 )}

拡張

x^{4} (x^{9} + 15 x^{6} + 75 x^{3} + 125) : x^{13} + 15 x^{10} + 75 x^{7} + 125 x^{4}

= \frac{( x ^{10} + 15 x ^{8} + 90 x ^{6} + 270 x ^{4} + 405 x ^{2} + 243 ) ( x ^{2} - 2 x + 1 )}{x ^{13} + 15 x ^{10} + 75 x ^{7} + 125 x ^{4}}

拡張

(x^{10} + 15 x^{8} + 90 x^{6} + 270 x^{4} + 405 x^{2} + 243) (x^{2} - 2 x + 1) : x^{12} - 2 x^{11} + 16 x^{10} - 30 x^{9} + 105 x^{8} - 180 x^{7} + 360 x^{6} - 540 x^{5} + 675 x^{4} - 810 x^{3} + 648 x^{2} - 486 x + 243

= \frac{x ^{12} - 2 x ^{11} + 16 x ^{10} - 30 x ^{9} + 105 x ^{8} - 180 x ^{7} + 360 x ^{6} - 540 x ^{5} + 675 x ^{4} - 810 x ^{3} + 648 x ^{2} - 486 x + 243}{x ^{13} + 15 x ^{10} + 75 x ^{7} + 125 x ^{4}}