Представьте выражение в виде произведения многочленов: a) (x - 7)8 - 1; г) (5а - 4)3 - 8; ж) 64(m + 5)3 – т³; б) (y + 9)³ + 27; д) (76 + 3)3 + 125; 3) п³(n - 8)3 + 8n6; в) 64- (2-5); e) 0,216 + (c + 0,2)³; и) 343r6 - (r – 11)³.

Ответы:

1. a) \[(x - 7)^3 - 1 = (x - 7 - 1)((x - 7)^2 + (x - 7) + 1) = (x - 8)(x^2 - 14x + 49 + x - 7 + 1) = (x - 8)(x^2 - 13x + 43)\]

2. б) \[(y + 9)^3 + 27 = (y + 9)^3 + 3^3 = (y + 9 + 3)((y + 9)^2 - 3(y + 9) + 9) = (y + 12)(y^2 + 18y + 81 - 3y - 27 + 9) = (y + 12)(y^2 + 15y + 63)\]

3. в) \[64 - (z - 5)^3 = 4^3 - (z - 5)^3 = (4 - (z - 5))(16 + 4(z - 5) + (z - 5)^2) = (4 - z + 5)(16 + 4z - 20 + z^2 - 10z + 25) = (9 - z)(z^2 - 6z + 21)\]

4. г) \[(5a - 4)^3 - 8 = (5a - 4)^3 - 2^3 = (5a - 4 - 2)((5a - 4)^2 + 2(5a - 4) + 4) = (5a - 6)(25a^2 - 40a + 16 + 10a - 8 + 4) = (5a - 6)(25a^2 - 30a + 12)\]

5. д) \[(7b + 3)^3 + 125 = (7b + 3)^3 + 5^3 = (7b + 3 + 5)((7b + 3)^2 - 5(7b + 3) + 25) = (7b + 8)(49b^2 + 42b + 9 - 35b - 15 + 25) = (7b + 8)(49b^2 + 7b + 19)\]

6. e) \[0.216 + (c + 0.2)^3 = (0.6)^3 + (c + 0.2)^3 = (0.6 + c + 0.2)((0.6)^2 - 0.6(c + 0.2) + (c + 0.2)^2) = (c + 0.8)(0.36 - 0.6c - 0.12 + c^2 + 0.4c + 0.04) = (c + 0.8)(c^2 - 0.2c + 0.28)\]

7. ж) \[64(m + 5)^3 - m^3 = (4(m + 5))^3 - m^3 = (4m + 20 - m)((4m + 20)^2 + (4m + 20)m + m^2) = (3m + 20)(16m^2 + 160m + 400 + 4m^2 + 20m + m^2) = (3m + 20)(21m^2 + 180m + 400)\]

8. з) \[n^3(n - 8)^3 + 8n^6 = n^3((n - 8)^3 + 8n^3) = n^3((n - 8 + 2n)((n - 8)^2 - 2n(n - 8) + 4n^2)) = n^3((3n - 8)(n^2 - 16n + 64 - 2n^2 + 16n + 4n^2)) = n^3((3n - 8)(3n^2 + 64))\]

9. и) \[343r^6 - (r - 11)^3 = (7r^2)^3 - (r - 11)^3 = (7r^2 - (r - 11))((7r^2)^2 + 7r^2(r - 11) + (r - 11)^2) = (7r^2 - r + 11)(49r^4 + 7r^3 - 77r^2 + r^2 - 22r + 121) = (7r^2 - r + 11)(49r^4 + 7r^3 - 76r^2 - 22r + 121)\]

Ответы: