확장 ((x+3i)^5*(3-4i))/(16)

해법

확장

\frac{( x + 3 i ) ^{5} \cdot ( 3 - 4 i )}{16}

\frac{4 ( 243 + 15 x ^{4} ) - 1080 x ^{2} + 1215 x - 270 x ^{3} + 3 x ^{5}}{16} + i \frac{729 + 4 ( - x ^{5} - 405 x ) + 360 x ^{3} + 45 x ^{4} - 810 x ^{2}}{16}

솔루션 단계

\frac{( x + 3 i ) ^{5} ( 3 - 4 i )}{16}

(x + 3 i)^{5} = x^{5} + 15 i x^{4} - 90 x^{3} - 270 i x^{2} + 405 x + 243 i

= \frac{( 3 - 4 i ) ( x ^{5} + 15 i x ^{4} - 90 x ^{3} - 270 i x ^{2} + 405 x + 243 i )}{16}

(x^{5} + 15 i x^{4} - 90 x^{3} - 270 i x^{2} + 405 x + 243 i) (3 - 4 i) : 3 (x^{5} - 90 x^{3} + 405 x) + 4 (15 x^{4} - 270 x^{2} + 243) + i (3 (15 x^{4} - 270 x^{2} + 243) - 4 (x^{5} - 90 x^{3} + 405 x))

= \frac{3 ( x ^{5} - 90 x ^{3} + 405 x ) + 4 ( 15 x ^{4} - 270 x ^{2} + 243 ) + i ( 3 ( 15 x ^{4} - 270 x ^{2} + 243 ) - 4 ( x ^{5} - 90 x ^{3} + 405 x ) )}{16}

- 4 (x^{5} - 90 x^{3} + 405 x) + 3 (15 x^{4} - 270 x^{2} + 243)

확대한다:

- 4 x^{5} + 360 x^{3} - 1620 x + 45 x^{4} - 810 x^{2} + 729

= \frac{3 ( x ^{5} - 90 x ^{3} + 405 x ) + 4 ( 15 x ^{4} - 270 x ^{2} + 243 ) + i ( - 4 x ^{5} + 45 x ^{4} + 360 x ^{3} - 810 x ^{2} - 1620 x + 729 )}{16}

3 (x^{5} - 90 x^{3} + 405 x) + 4 (15 x^{4} - 270 x^{2} + 243) + (- 4 x^{5} + 360 x^{3} - 1620 x + 45 x^{4} - 810 x^{2} + 729) i

확대한다:

3 x^{5} - 270 x^{3} + 1215 x + 60 x^{4} - 1080 x^{2} + 972 - 4 i x^{5} + 360 i x^{3} - 1620 i x + 45 i x^{4} - 810 i x^{2} + 729 i

= \frac{3 x ^{5} - 270 x ^{3} + 1215 x + 60 x ^{4} - 1080 x ^{2} + 972 - 4 i x ^{5} + 360 i x ^{3} - 1620 i x + 45 i x ^{4} - 810 i x ^{2} + 729 i}{16}

분수 규칙 적용:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{3 x ^{5} - 270 x ^{3} + 1215 x + 60 x ^{4} - 1080 x ^{2} + 972 - 4 i x ^{5} + 360 i x ^{3} - 1620 i x + 45 i x ^{4} - 810 i x ^{2} + 729 i}{16} = \frac{3 x ^{5}}{16} - \frac{270 x ^{3}}{16} + \frac{1215 x}{16} + \frac{60 x ^{4}}{16} - \frac{1080 x ^{2}}{16} + \frac{972}{16} - \frac{4 i x ^{5}}{16} + \frac{360 i x ^{3}}{16} - \frac{1620 i x}{16} + \frac{45 i x ^{4}}{16} - \frac{810 i x ^{2}}{16} + \frac{729 i}{16}

= \frac{3 x ^{5}}{16} - \frac{270 x ^{3}}{16} + \frac{1215 x}{16} + \frac{60 x ^{4}}{16} - \frac{1080 x ^{2}}{16} + \frac{972}{16} - \frac{4 i x ^{5}}{16} + \frac{360 i x ^{3}}{16} - \frac{1620 i x}{16} + \frac{45 i x ^{4}}{16} - \frac{810 i x ^{2}}{16} + \frac{729 i}{16}

\frac{270 x ^{3}}{16} = \frac{135 x ^{3}}{8}

\frac{60 x ^{4}}{16} = \frac{15 x ^{4}}{4}

\frac{1080 x ^{2}}{16} = \frac{135 x ^{2}}{2}

\frac{972}{16} = \frac{243}{4}

\frac{4 i x ^{5}}{16} = \frac{i x ^{5}}{4}

\frac{360 i x ^{3}}{16} = \frac{45 i x ^{3}}{2}

\frac{1620 i x}{16} = \frac{405 i x}{4}

\frac{810 i x ^{2}}{16} = \frac{405 i x ^{2}}{8}

= \frac{3 x ^{5}}{16} - \frac{135 x ^{3}}{8} + \frac{1215 x}{16} + \frac{15 x ^{4}}{4} - \frac{135 x ^{2}}{2} + \frac{243}{4} - \frac{i x ^{5}}{4} + \frac{45 i x ^{3}}{2} - \frac{405 i x}{4} + \frac{45 i x ^{4}}{16} - \frac{405 i x ^{2}}{8} + \frac{729 i}{16}

다시 쓰다

\frac{3 x ^{5}}{16} - \frac{135 x ^{3}}{8} + \frac{1215 x}{16} + \frac{15 x ^{4}}{4} - \frac{135 x ^{2}}{2} + \frac{243}{4} - \frac{i x ^{5}}{4} + \frac{45 i x ^{3}}{2} - \frac{405 i x}{4} + \frac{45 i x ^{4}}{16} - \frac{405 i x ^{2}}{8} + \frac{729 i}{16}

표준복합형태로:

\frac{3 x ^{5} + 1215 x - 270 x ^{3} + 4 ( 15 x ^{4} + 243 ) - 1080 x ^{2}}{16} + \frac{4 ( - x ^{5} - 405 x ) + 360 x ^{3} + 45 x ^{4} + 729 - 810 x ^{2}}{16} i

= \frac{3 x ^{5} + 1215 x - 270 x ^{3} + 4 ( 15 x ^{4} + 243 ) - 1080 x ^{2}}{16} + \frac{4 ( - x ^{5} - 405 x ) + 360 x ^{3} + 45 x ^{4} + 729 - 810 x ^{2}}{16} i