확장 ((x+i)^9)/((x-i)^7)

해법

확장

\frac{( x + i ) ^{9}}{( x - i ) ^{7}}

\frac{1 + x ^{16} - 120 x ^{14} - 120 x ^{2} + 12870 x ^{8} + 1820 x ^{12} + 1820 x ^{4} - 8008 x ^{10} - 8008 x ^{6}}{1 + x ^{14} + 21 x ^{10} + 21 x ^{4} + 35 x ^{6} + 35 x ^{8} + 7 x ^{12} + 7 x ^{2}} + i \frac{11440 x ^{7} - 11440 x ^{9} - 16 x + 16 x ^{15} + 4368 x ^{11} - 4368 x ^{5} - 560 x ^{13} + 560 x ^{3}}{1 + x ^{14} + 21 x ^{10} + 21 x ^{4} + 35 x ^{6} + 35 x ^{8} + 7 x ^{12} + 7 x ^{2}}

솔루션 단계

\frac{( x + i ) ^{9}}{( x - i ) ^{7}}

(x - i)^{7} = x^{7} - 7 i x^{6} - 21 x^{5} + 35 i x^{4} + 35 x^{3} - 21 i x^{2} - 7 x + i

= \frac{( x + i ) ^{9}}{x ^{7} - 7 i x ^{6} - 21 x ^{5} + 35 i x ^{4} + 35 x ^{3} - 21 i x ^{2} - 7 x + i}

(x + i)^{9} = x^{9} + 9 i x^{8} - 36 x^{7} - 84 i x^{6} + 126 x^{5} + 126 i x^{4} - 84 x^{3} - 36 i x^{2} + 9 x + i

= \frac{x ^{9} + 9 i x ^{8} - 36 x ^{7} - 84 i x ^{6} + 126 x ^{5} + 126 i x ^{4} - 84 x ^{3} - 36 i x ^{2} + 9 x + i}{x ^{7} - 7 i x ^{6} - 21 x ^{5} + 35 i x ^{4} + 35 x ^{3} - 21 i x ^{2} - 7 x + i}

\frac{x ^{9} + 9 i x ^{8} - 36 x ^{7} - 84 i x ^{6} + 126 x ^{5} + 126 i x ^{4} - 84 x ^{3} - 36 i x ^{2} + 9 x + i}{x ^{7} - 7 i x ^{6} - 21 x ^{5} + 35 i x ^{4} + 35 x ^{3} - 21 i x ^{2} - 7 x + i}

합리화합니다 :

\frac{x ^{16} - 120 x ^{14} + 1820 x ^{12} - 8008 x ^{10} + 12870 x ^{8} - 8008 x ^{6} + 1820 x ^{4} - 120 x ^{2} + i ( 16 x ^{15} - 560 x ^{13} + 4368 x ^{11} - 11440 x ^{9} + 11440 x ^{7} - 4368 x ^{5} + 560 x ^{3} - 16 x ) + 1}{x ^{14} + 7 x ^{12} + 21 x ^{10} + 35 x ^{8} + 35 x ^{6} + 21 x ^{4} + 7 x ^{2} + 1}

= \frac{x ^{16} - 120 x ^{14} + 1820 x ^{12} - 8008 x ^{10} + 12870 x ^{8} - 8008 x ^{6} + 1820 x ^{4} - 120 x ^{2} + 1 + i ( 16 x ^{15} - 560 x ^{13} + 4368 x ^{11} - 11440 x ^{9} + 11440 x ^{7} - 4368 x ^{5} + 560 x ^{3} - 16 x )}{x ^{14} + 7 x ^{12} + 21 x ^{10} + 35 x ^{8} + 35 x ^{6} + 21 x ^{4} + 7 x ^{2} + 1}

다시 쓰다

\frac{x ^{16} - 120 x ^{14} + 1820 x ^{12} - 8008 x ^{10} + 12870 x ^{8} - 8008 x ^{6} + 1820 x ^{4} - 120 x ^{2} + 1 + i ( 16 x ^{15} - 560 x ^{13} + 4368 x ^{11} - 11440 x ^{9} + 11440 x ^{7} - 4368 x ^{5} + 560 x ^{3} - 16 x )}{x ^{14} + 7 x ^{12} + 21 x ^{10} + 35 x ^{8} + 35 x ^{6} + 21 x ^{4} + 7 x ^{2} + 1}

표준복합형태로:

\frac{x ^{16} - 120 x ^{14} + 1820 x ^{12} - 8008 x ^{10} + 12870 x ^{8} - 8008 x ^{6} + 1820 x ^{4} - 120 x ^{2} + 1}{x ^{14} + 7 x ^{12} + 21 x ^{10} + 35 x ^{8} + 35 x ^{6} + 21 x ^{4} + 7 x ^{2} + 1} + \frac{16 x ^{15} - 560 x ^{13} + 4368 x ^{11} - 11440 x ^{9} + 11440 x ^{7} - 4368 x ^{5} + 560 x ^{3} - 16 x}{x ^{14} + 7 x ^{12} + 21 x ^{10} + 35 x ^{8} + 35 x ^{6} + 21 x ^{4} + 7 x ^{2} + 1} i

= \frac{x ^{16} - 120 x ^{14} + 1820 x ^{12} - 8008 x ^{10} + 12870 x ^{8} - 8008 x ^{6} + 1820 x ^{4} - 120 x ^{2} + 1}{x ^{14} + 7 x ^{12} + 21 x ^{10} + 35 x ^{8} + 35 x ^{6} + 21 x ^{4} + 7 x ^{2} + 1} + \frac{16 x ^{15} - 560 x ^{13} + 4368 x ^{11} - 11440 x ^{9} + 11440 x ^{7} - 4368 x ^{5} + 560 x ^{3} - 16 x}{x ^{14} + 7 x ^{12} + 21 x ^{10} + 35 x ^{8} + 35 x ^{6} + 21 x ^{4} + 7 x ^{2} + 1} i