因数 (a+b)^3-1000c^3

解

因数

(a + b)^{3} - 1000 c^{3}

(a + b - 10 c) ((a + b)^{2} + 10 c (a + b) + 100 c^{2})

解答ステップ

(a + b)^{3} - 1000 c^{3}

(a + b)^{3} - 1000 c^{3}

を書き換え

(a + b)^{3} - (10 c)^{3}

= (a + b)^{3} - (10 c)^{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} - (10 c)^{3} = ((a + b) - 10 c) ((a + b)^{2} + 10 (a + b) c + 1 0^{2} c^{2})

= ((a + b) - 10 c) ((a + b)^{2} + 10 c (a + b) + 1 0^{2} c^{2})

改良

= (a + b - 10 c) (100 c^{2} + 10 c (a + b) + (a + b)^{2})