12. Запишите многочлен в виде произведения:

Решение:

1. \( 4x^3 - 4y^3 = 4(x^3 - y^3) = 4(x - y)(x^2 + xy + y^2) \)
2. \( 7a^2 + 7b^3 = 7(a^2 + b^3) \)
3. \( 1000a^3m^3 - a^3n^3 = a^3(1000m^3 - n^3) = a^3(10m - n)(100m^2 + 10mn + n^2) \)
4. \( 16x^2 - 2 = 2(8x^2 - 1) \)
5. \( 1000m + m^4 = m(1000 + m^3) = m(10 + m)(100 - 10m + m^2) \)
6. \( x^5 - x^2 = x^2(x^3 - 1) = x^2(x - 1)(x^2 + x + 1) \)
7. \( y^3 + y^6 = y^3(1 + y^3) = y^3(1 + y)(1 - y + y^2) \)
8. \( 27m^2 - m^5 = m^2(27 - m^3) = m^2(3 - m)(9 + 3m + m^2) \)

Ответ:

1. \( 4(x - y)(x^2 + xy + y^2) \)
2. \( 7(a^2 + b^3) \)
3. \( a^3(10m - n)(100m^2 + 10mn + n^2) \)
4. \( 2(8x^2 - 1) \)
5. \( m(10 + m)(100 - 10m + m^2) \)
6. \( x^2(x - 1)(x^2 + x + 1) \)
7. \( y^3(1 + y)(1 - y + y^2) \)
8. \( m^2(3 - m)(9 + 3m + m^2) \)