因数 125(x+2)^3+64(x-5)^3

解

因数

125 (x + 2)^{3} + 64 (x - 5)^{3}

7 (9 x - 10) (3 x^{2} + 100)

解答ステップ

125 (x + 2)^{3} + 64 (x - 5)^{3}

125 (x + 2)^{3} + 64 (x - 5)^{3}

を書き換え

(5 (x + 2))^{3} + (4 (x - 5))^{3}

= (5 (x + 2))^{3} + (4 (x - 5))^{3}

立方数の和の公式を適用する:

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

(5 (x + 2))^{3} + (4 (x - 5))^{3} = (5 (x + 2) + 4 (x - 5)) (5^{2} (x + 2)^{2} - 5 \cdot 4 (x - 5) (x + 2) + 4^{2} (x - 5)^{2})

= (4 (x - 5) + 5 (x + 2)) (4^{2} (x - 5)^{2} - 5 \cdot 4 (x - 5) (x + 2) + 5^{2} (x + 2)^{2})

改良

= (4 (x - 5) + 5 (x + 2)) (16 (x - 5)^{2} - 20 (x - 5) (x + 2) + 25 (x + 2)^{2})

因数

25 (x + 2)^{2} - 20 (x - 5) (x + 2) + 16 (x - 5)^{2} : 7 (3 x^{2} + 100)

= 7 (4 (x - 5) + 5 (x + 2)) (3 x^{2} + 100)

拡張

4 (x - 5) + 5 (x + 2) : 9 x - 10

= 7 (9 x - 10) (3 x^{2} + 100)