展開する ((a+b+c)(-a+b+c)(a-b+c)(a+b-c))/(16)

解

展開する

\frac{( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c )}{16}

- \frac{a ^{4}}{16} + \frac{a ^{2} b ^{2}}{8} + \frac{a ^{2} c ^{2}}{8} - \frac{b ^{4}}{16} - \frac{c ^{4}}{16} + \frac{b ^{2} c ^{2}}{8}

解答ステップ

\frac{( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c )}{16}

拡張

(a + b + c) (- a + b + c) (a - b + c) (a + b - c) : - a^{4} + 2 a^{2} b^{2} + 2 a^{2} c^{2} - b^{4} - c^{4} + 2 b^{2} c^{2}

= \frac{- a ^{4} + 2 a ^{2} b ^{2} + 2 a ^{2} c ^{2} - b ^{4} - c ^{4} + 2 b ^{2} c ^{2}}{16}

分数の規則を適用する:

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

\frac{- a ^{4} + 2 a ^{2} b ^{2} + 2 a ^{2} c ^{2} - b ^{4} - c ^{4} + 2 b ^{2} c ^{2}}{16} = - \frac{a ^{4}}{16} + \frac{2 a ^{2} b ^{2}}{16} + \frac{2 a ^{2} c ^{2}}{16} - \frac{b ^{4}}{16} - \frac{c ^{4}}{16} + \frac{2 b ^{2} c ^{2}}{16}

= - \frac{a ^{4}}{16} + \frac{2 a ^{2} b ^{2}}{16} + \frac{2 a ^{2} c ^{2}}{16} - \frac{b ^{4}}{16} - \frac{c ^{4}}{16} + \frac{2 b ^{2} c ^{2}}{16}

キャンセル

\frac{2 a ^{2} b ^{2}}{16} : \frac{a ^{2} b ^{2}}{8}

= - \frac{a ^{4}}{16} + \frac{a ^{2} b ^{2}}{8} + \frac{2 a ^{2} c ^{2}}{16} - \frac{b ^{4}}{16} - \frac{c ^{4}}{16} + \frac{2 b ^{2} c ^{2}}{16}

キャンセル

\frac{2 a ^{2} c ^{2}}{16} : \frac{a ^{2} c ^{2}}{8}

= - \frac{a ^{4}}{16} + \frac{a ^{2} b ^{2}}{8} + \frac{a ^{2} c ^{2}}{8} - \frac{b ^{4}}{16} - \frac{c ^{4}}{16} + \frac{2 b ^{2} c ^{2}}{16}

キャンセル

\frac{2 b ^{2} c ^{2}}{16} : \frac{b ^{2} c ^{2}}{8}

= - \frac{a ^{4}}{16} + \frac{a ^{2} b ^{2}}{8} + \frac{a ^{2} c ^{2}}{8} - \frac{b ^{4}}{16} - \frac{c ^{4}}{16} + \frac{b ^{2} c ^{2}}{8}