因数 (a+b)^3-(2a+3b)^3

解

因数

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

(- a - 2 b) (7 a^{2} + 19 ab + 13 b^{2})

解答ステップ

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

立方数の差の公式を適用する:

x^{3} - y^{3} = (x - y) (x^{2} + x y + y^{2})

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

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

改良

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

拡張

(a + b)^{2} + (a + b) (2 a + 3 b) + (2 a + 3 b)^{2} : 7 a^{2} + 19 ab + 13 b^{2}

= (- a - 2 b) (7 a^{2} + 19 ab + 13 b^{2})