展開する (a^2(b-a)^2(n+1)(2n+1))/(n^2)

解

展開する

\frac{a ^{2} ( b - a ) ^{2} ( n + 1 ) ( 2 n + 1 )}{n ^{2}}

2 a^{2} b^{2} + \frac{3 a ^{2} b ^{2}}{n} + \frac{a ^{2} b ^{2}}{n ^{2}} - 4 a^{3} b - \frac{6 a ^{3} b}{n} - \frac{2 a ^{3} b}{n ^{2}} + 2 a^{4} + \frac{3 a ^{4}}{n} + \frac{a ^{4}}{n ^{2}}

解答ステップ

\frac{a ^{2} ( b - a ) ^{2} ( n + 1 ) ( 2 n + 1 )}{n ^{2}}

(b - a)^{2} = b^{2} - 2 ab + a^{2}

= \frac{a ^{2} ( a ^{2} - 2 ab + b ^{2} ) ( n + 1 ) ( 2 n + 1 )}{n ^{2}}

拡張

a^{2} (b^{2} - 2 ba + a^{2}) (n + 1) (2 n + 1) : 2 n^{2} a^{2} b^{2} + 3 n a^{2} b^{2} + a^{2} b^{2} - 4 n^{2} a^{3} b - 6 n a^{3} b - 2 a^{3} b + 2 n^{2} a^{4} + 3 n a^{4} + a^{4}

= \frac{2 n ^{2} a ^{2} b ^{2} + 3 n a ^{2} b ^{2} + a ^{2} b ^{2} - 4 n ^{2} a ^{3} b - 6 n a ^{3} b - 2 a ^{3} b + 2 n ^{2} a ^{4} + 3 n a ^{4} + a ^{4}}{n ^{2}}

分数の規則を適用する:

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

\frac{2 n ^{2} a ^{2} b ^{2} + 3 n a ^{2} b ^{2} + a ^{2} b ^{2} - 4 n ^{2} a ^{3} b - 6 n a ^{3} b - 2 a ^{3} b + 2 n ^{2} a ^{4} + 3 n a ^{4} + a ^{4}}{n ^{2}} = \frac{2 n ^{2} a ^{2} b ^{2}}{n ^{2}} + \frac{3 n a ^{2} b ^{2}}{n ^{2}} + \frac{a ^{2} b ^{2}}{n ^{2}} - \frac{4 n ^{2} a ^{3} b}{n ^{2}} - \frac{6 n a ^{3} b}{n ^{2}} - \frac{2 a ^{3} b}{n ^{2}} + \frac{2 n ^{2} a ^{4}}{n ^{2}} + \frac{3 n a ^{4}}{n ^{2}} + \frac{a ^{4}}{n ^{2}}

= \frac{2 n ^{2} a ^{2} b ^{2}}{n ^{2}} + \frac{3 n a ^{2} b ^{2}}{n ^{2}} + \frac{a ^{2} b ^{2}}{n ^{2}} - \frac{4 n ^{2} a ^{3} b}{n ^{2}} - \frac{6 n a ^{3} b}{n ^{2}} - \frac{2 a ^{3} b}{n ^{2}} + \frac{2 n ^{2} a ^{4}}{n ^{2}} + \frac{3 n a ^{4}}{n ^{2}} + \frac{a ^{4}}{n ^{2}}

\frac{2 n ^{2} a ^{2} b ^{2}}{n ^{2}} = 2 a^{2} b^{2}

\frac{3 n a ^{2} b ^{2}}{n ^{2}} = \frac{3 a ^{2} b ^{2}}{n}

\frac{4 n ^{2} a ^{3} b}{n ^{2}} = 4 a^{3} b

\frac{6 n a ^{3} b}{n ^{2}} = \frac{6 a ^{3} b}{n}

\frac{2 n ^{2} a ^{4}}{n ^{2}} = 2 a^{4}

\frac{3 n a ^{4}}{n ^{2}} = \frac{3 a ^{4}}{n}

= 2 a^{2} b^{2} + \frac{3 a ^{2} b ^{2}}{n} + \frac{a ^{2} b ^{2}}{n ^{2}} - 4 a^{3} b - \frac{6 a ^{3} b}{n} - \frac{2 a ^{3} b}{n ^{2}} + 2 a^{4} + \frac{3 a ^{4}}{n} + \frac{a ^{4}}{n ^{2}}