III. Сумма и разность кубов. a³ ± b³ = (a ± b) (a² ∓ ab + b²) 5. Разложите на множители. 1) x³ + y³ 2) m³ – n³ 3) a³ + 8 4) b³ – 27 5) 64 + c³ 6) 125 – d³ 7) m³ + 216 8) 343 + n³ 9) 8x³ – 1 10) 1 + 1000y³ 11) 16a⁶ – 27c³ 12) 343b³ + 8d³ 13) 64x⁶ – 125m³ 14) 729n¹² + 1 15) 8y³ + 512z⁹

1. \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)
2. \(m^3 - n^3 = (m - n)(m^2 + mn + n^2)\)
3. \(a^3 + 8 = a^3 + 2^3 = (a + 2)(a^2 - 2a + 4)\)
4. \(b^3 - 27 = b^3 - 3^3 = (b - 3)(b^2 + 3b + 9)\)
5. \(64 + c^3 = 4^3 + c^3 = (4 + c)(16 - 4c + c^2)\)
6. \(125 - d^3 = 5^3 - d^3 = (5 - d)(25 + 5d + d^2)\)
7. \(m^3 + 216 = m^3 + 6^3 = (m + 6)(m^2 - 6m + 36)\)
8. \(343 + n^3 = 7^3 + n^3 = (7 + n)(49 - 7n + n^2)\)
9. \(8x^3 - 1 = (2x)^3 - 1^3 = (2x - 1)(4x^2 + 2x + 1)\)
10. \(1 + 1000y^3 = 1^3 + (10y)^3 = (1 + 10y)(1 - 10y + 100y^2)\)
11. \(16a^6 - 27c^3 = (2a^2)^3 - (3c)^3 = (2a^2 - 3c)(4a^4 + 6a^2c + 9c^2)\)
12. \(343b^3 + 8d^3 = (7b)^3 + (2d)^3 = (7b + 2d)(49b^2 - 14bd + 4d^2)\)
13. \(64x^6 - 125m^3 = (4x^2)^3 - (5m)^3 = (4x^2 - 5m)(16x^4 + 20x^2m + 25m^2)\)
14. \(729n^{12} + 1 = (9n^4)^3 + 1^3 = (9n^4 + 1)(81n^8 - 9n^4 + 1)\)
15. \(8y^3 + 512z^9 = (2y)^3 + (8z^3)^3 = (2y + 8z^3)(4y^2 - 16yz^3 + 64z^6)\)

Ответ:

• \((x + y)(x^2 - xy + y^2)\)
• \((m - n)(m^2 + mn + n^2)\)
• \((a + 2)(a^2 - 2a + 4)\)
• \((b - 3)(b^2 + 3b + 9)\)
• \((4 + c)(16 - 4c + c^2)\)
• \((5 - d)(25 + 5d + d^2)\)
• \((m + 6)(m^2 - 6m + 36)\)
• \((7 + n)(49 - 7n + n^2)\)
• \((2x - 1)(4x^2 + 2x + 1)\)
• \((1 + 10y)(1 - 10y + 100y^2)\)
• \((2a^2 - 3c)(4a^4 + 6a^2c + 9c^2)\)
• \((7b + 2d)(49b^2 - 14bd + 4d^2)\)
• \((4x^2 - 5m)(16x^4 + 20x^2m + 25m^2)\)
• \((9n^4 + 1)(81n^8 - 9n^4 + 1)\)
• \((2y + 8z^3)(4y^2 - 16yz^3 + 64z^6)\)