因数 (a+b)^3-(30+4b)^3

解

因数

(a + b)^{3} - (30 + 4 b)^{3}

(- 3 b + a - 30) (21 b^{2} + 6 ab + 270 b + a^{2} + 30 a + 900)

解答ステップ

(a + b)^{3} - (30 + 4 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} - (30 + 4 b)^{3} = ((a + b) - (30 + 4 b)) ((a + b)^{2} + (30 + 4 b) (a + b) + (30 + 4 b)^{2})

= ((a + b) - (30 + 4 b)) ((30 + 4 b)^{2} + (30 + 4 b) (a + b) + (a + b)^{2})

改良

= (a - 3 b - 30) ((4 b + 30)^{2} + (4 b + 30) (a + b) + (a + b)^{2})

(30 + 4 b) (a + b) = 2 (2 b + 15) (a + b)

(30 + 4 b)^{2} = 4 (2 b + 15)^{2}

= (a - 3 b - 30) (2 (2 b + 15) (a + b) + 4 (2 b + 15)^{2} + (a + b)^{2})

拡張

(a + b)^{2} + 2 (2 b + 15) (a + b) + 4 (2 b + 15)^{2} : 21 b^{2} + 6 ab + 270 b + a^{2} + 30 a + 900

= (a - 3 b - 30) (a^{2} + 6 ab + 30 a + 21 b^{2} + 270 b + 900)