展開する (a+b+c)^4

解

展開する

(a + b + c)^{4}

a^{4} + 4 a^{3} b + 4 a^{3} c + 4 a c^{3} + 4 b c^{3} + 4 a b^{3} + 4 b^{3} c + 6 a^{2} b^{2} + 6 a^{2} c^{2} + 6 b^{2} c^{2} + c^{4} + 12 a^{2} b c + 12 ab c^{2} + 12 a b^{2} c + b^{4}

解答ステップ

(a + b + c)^{4}

簡素化

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

= (a + b + c) (a + b + c) (a + b + c) (a + b + c)

拡張

(a + b + c) (a + b + c) : a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2}

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

拡張

(a^{2} + 2 ab + 2 a c + b^{2} + 2 b c + c^{2}) (a + b + c) : a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + c^{3} + 3 b^{2} c + 6 ab c + b^{3}

= (a^{3} + 3 a^{2} b + 3 a^{2} c + 3 a c^{2} + 3 b c^{2} + 3 a b^{2} + c^{3} + 3 b^{2} c + 6 ab c + b^{3}) (a + b + c)

括弧を分配する

= a^{3} a + a^{3} b + a^{3} c + 3 a^{2} ba + 3 a^{2} bb + 3 a^{2} b c + 3 a^{2} c a + 3 a^{2} c b + 3 a^{2} cc + 3 a c^{2} a + 3 a c^{2} b + 3 a c^{2} c + 3 b c^{2} a + 3 b c^{2} b + 3 b c^{2} c + 3 a b^{2} a + 3 a b^{2} b + 3 a b^{2} c + c^{3} a + c^{3} b + c^{3} c + 3 b^{2} c a + 3 b^{2} c b + 3 b^{2} cc + 6 ab c a + 6 ab c b + 6 ab cc + b^{3} a + b^{3} b + b^{3} c

簡素化

a^{3} a + a^{3} b + a^{3} c + 3 a^{2} ba + 3 a^{2} bb + 3 a^{2} b c + 3 a^{2} c a + 3 a^{2} c b + 3 a^{2} cc + 3 a c^{2} a + 3 a c^{2} b + 3 a c^{2} c + 3 b c^{2} a + 3 b c^{2} b + 3 b c^{2} c + 3 a b^{2} a + 3 a b^{2} b + 3 a b^{2} c + c^{3} a + c^{3} b + c^{3} c + 3 b^{2} c a + 3 b^{2} c b + 3 b^{2} cc + 6 ab c a + 6 ab c b + 6 ab cc + b^{3} a + b^{3} b + b^{3} c : a^{4} + 4 a^{3} b + 4 a^{3} c + 4 a c^{3} + 4 b c^{3} + 4 a b^{3} + 4 b^{3} c + 6 a^{2} b^{2} + 6 a^{2} c^{2} + 6 b^{2} c^{2} + c^{4} + 12 a^{2} b c + 12 ab c^{2} + 12 a b^{2} c + b^{4}

= a^{4} + 4 a^{3} b + 4 a^{3} c + 4 a c^{3} + 4 b c^{3} + 4 a b^{3} + 4 b^{3} c + 6 a^{2} b^{2} + 6 a^{2} c^{2} + 6 b^{2} c^{2} + c^{4} + 12 a^{2} b c + 12 ab c^{2} + 12 a b^{2} c + b^{4}