展開する (9(x+2)^3(x-6))/(2(x-7)^{1/2)}

解

展開する

\frac{9 ( x + 2 ) ^{3} ( x - 6 )}{2 ( x - 7 ) ^{\frac{1}{2}}}

\frac{9 x ^{4}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{108 x ^{2}}{( x - 7 ) ^{\frac{1}{2}}} - \frac{288 x}{( x - 7 ) ^{\frac{1}{2}}} - \frac{216}{( x - 7 ) ^{\frac{1}{2}}}

解答ステップ

\frac{9 ( x + 2 ) ^{3} ( x - 6 )}{2 ( x - 7 ) ^{\frac{1}{2}}}

(x + 2)^{3} = x^{3} + 6 x^{2} + 12 x + 8

= \frac{9 ( x ^{3} + 6 x ^{2} + 12 x + 8 ) ( x - 6 )}{2 ( x - 7 ) ^{\frac{1}{2}}}

拡張

9 (x^{3} + 6 x^{2} + 12 x + 8) (x - 6) : 9 x^{4} - 216 x^{2} - 576 x - 432

= \frac{9 x ^{4} - 216 x ^{2} - 576 x - 432}{2 ( x - 7 ) ^{\frac{1}{2}}}

分数の規則を適用する:

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

\frac{9 x ^{4} - 216 x ^{2} - 576 x - 432}{2 ( x - 7 ) ^{\frac{1}{2}}} = \frac{9 x ^{4}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{216 x ^{2}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{576 x}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{432}{2 ( x - 7 ) ^{\frac{1}{2}}}

= \frac{9 x ^{4}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{216 x ^{2}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{576 x}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{432}{2 ( x - 7 ) ^{\frac{1}{2}}}

数を割る:

\frac{216}{2} = 108

= \frac{9 x ^{4}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{108 x ^{2}}{( x - 7 ) ^{\frac{1}{2}}} - \frac{576 x}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{432}{2 ( x - 7 ) ^{\frac{1}{2}}}

数を割る:

\frac{576}{2} = 288

= \frac{9 x ^{4}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{108 x ^{2}}{( x - 7 ) ^{\frac{1}{2}}} - \frac{288 x}{( x - 7 ) ^{\frac{1}{2}}} - \frac{432}{2 ( x - 7 ) ^{\frac{1}{2}}}

数を割る:

\frac{432}{2} = 216

= \frac{9 x ^{4}}{2 ( x - 7 ) ^{\frac{1}{2}}} - \frac{108 x ^{2}}{( x - 7 ) ^{\frac{1}{2}}} - \frac{288 x}{( x - 7 ) ^{\frac{1}{2}}} - \frac{216}{( x - 7 ) ^{\frac{1}{2}}}