簡約 (((a-b)^3+(b-c)^3+(c-a)^3))/((9(a-b)(b-c)(c-a)))

解

簡約

\frac{( ( a - b ) ^{3} + ( b - c ) ^{3} + ( c - a ) ^{3} )}{( 9 ( a - b ) ( b - c ) ( c - a ) )}

\frac{- a ^{2} b + a b ^{2} - b ^{2} c + b c ^{2} - a c ^{2} + a ^{2} c}{3 ( a - b ) ( b - c ) ( c - a )}

解答ステップ

\frac{( a - b ) ^{3} + ( b - c ) ^{3} + ( c - a ) ^{3}}{9 ( a - b ) ( b - c ) ( c - a )}

拡張

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

= \frac{- 3 a ^{2} b + 3 a b ^{2} - 3 b ^{2} c + 3 b c ^{2} - 3 a c ^{2} + 3 a ^{2} c}{9 ( a - b ) ( b - c ) ( c - a )}

共通項をくくり出す

3

= \frac{3 ( - a ^{2} b + a b ^{2} - b ^{2} c + b c ^{2} - a c ^{2} + a ^{2} c )}{9 ( a - b ) ( b - c ) ( c - a )}

数を因数に分解する:

9 = 3 \cdot 3

= \frac{3 ( - a ^{2} b + a b ^{2} - b ^{2} c + b c ^{2} - a c ^{2} + a ^{2} c )}{3 \cdot 3 ( a - b ) ( b - c ) ( c - a )}

共通因数を約分する:

3

= \frac{- a ^{2} b + a b ^{2} - b ^{2} c + b c ^{2} - a c ^{2} + a ^{2} c}{3 ( a - b ) ( b - c ) ( c - a )}