展開する ((5-x)^7(x-13)^2)/((x^2+7)^2)

解

展開する

\frac{( 5 - x ) ^{7} ( x - 13 ) ^{2}}{( x ^{2} + 7 ) ^{2}}

\frac{- x ^{9} + 61 x ^{8} - 1604 x ^{7} + 23940 x ^{6} - 224350 x ^{5} + 1373750 x ^{4} - 5512500 x ^{3} + 14012500 x ^{2} - 20515625 x + 13203125}{x ^{4} + 14 x ^{2} + 49}

解答ステップ

\frac{( 5 - x ) ^{7} ( x - 13 ) ^{2}}{( x ^{2} + 7 ) ^{2}}

(x^{2} + 7)^{2} = x^{4} + 14 x^{2} + 49

= \frac{( - x + 5 ) ^{7} ( x - 13 ) ^{2}}{x ^{4} + 14 x ^{2} + 49}

(5 - x)^{7} = 78125 - 109375 x + 65625 x^{2} - 21875 x^{3} + 4375 x^{4} - 525 x^{5} + 35 x^{6} - x^{7}

= \frac{( - x ^{7} + 35 x ^{6} - 525 x ^{5} + 4375 x ^{4} - 21875 x ^{3} + 65625 x ^{2} - 109375 x + 78125 ) ( x - 13 ) ^{2}}{x ^{4} + 14 x ^{2} + 49}

(x - 13)^{2} = x^{2} - 26 x + 169

= \frac{( - x ^{7} + 35 x ^{6} - 525 x ^{5} + 4375 x ^{4} - 21875 x ^{3} + 65625 x ^{2} - 109375 x + 78125 ) ( x ^{2} - 26 x + 169 )}{x ^{4} + 14 x ^{2} + 49}

拡張

(78125 - 109375 x + 65625 x^{2} - 21875 x^{3} + 4375 x^{4} - 525 x^{5} + 35 x^{6} - x^{7}) (x^{2} - 26 x + 169) : - x^{9} + 61 x^{8} - 1604 x^{7} + 23940 x^{6} - 224350 x^{5} + 1373750 x^{4} - 5512500 x^{3} + 14012500 x^{2} - 20515625 x + 13203125

= \frac{- x ^{9} + 61 x ^{8} - 1604 x ^{7} + 23940 x ^{6} - 224350 x ^{5} + 1373750 x ^{4} - 5512500 x ^{3} + 14012500 x ^{2} - 20515625 x + 13203125}{x ^{4} + 14 x ^{2} + 49}