2. Одночлены и многочлены.

2.1. Приведите к многочлену стандартного вида:

1. а) \( (4x^2 - 5x + 2) - (2 + 3x - x^2) \)
   \( = 4x^2 - 5x + 2 - 2 - 3x + x^2 \)
   \( = (4x^2 + x^2) + (-5x - 3x) + (2 - 2) \)
   \( = 5x^2 - 8x \)


2. б) \( (d + 2)(7 - d) \)
   \( = d \times 7 + d \times (-d) + 2 \times 7 + 2 \times (-d) \)
   \( = 7d - d^2 + 14 - 2d \)
   \( = -d^2 + (7d - 2d) + 14 \)
   \( = -d^2 + 5d + 14 \)


3. в) \( 2d(3d - 4) - 3d(3d - 1) \)
   \( = (2d \times 3d + 2d \times (-4)) - (3d \times 3d + 3d \times (-1)) \)
   \( = (6d^2 - 8d) - (9d^2 - 3d) \)
   \( = 6d^2 - 8d - 9d^2 + 3d \)
   \( = (6d^2 - 9d^2) + (-8d + 3d) \)
   \( = -3d^2 - 5d \)

2.2. Вычислите:

1. а) \( \frac{5^6 \cdot 125}{25^4} \)
   \( = \frac{5^6 \cdot 5^3}{(5^2)^4} \)
   \( = \frac{5^{6+3}}{5^{2 \times 4}} \)
   \( = \frac{5^9}{5^8} \)
   \( = 5^{9-8} = 5^1 = 5 \)


2. б) \( \frac{3^6 \cdot 216}{24^5} \)
   \( = \frac{3^6 \cdot (6^3)}{(3 \times 8)^5} \)
   \( = \frac{3^6 \times (2 \times 3)^3}{3^5 \times 8^5} \)
   \( = \frac{3^6 \times 2^3 \times 3^3}{3^5 \times (2^3)^5} \)
   \( = \frac{3^{6+3} \times 2^3}{3^5 \times 2^{15}} \)
   \( = \frac{3^9 \times 2^3}{3^5 \times 2^{15}} \)
   \( = 3^{9-5} \times 2^{3-15} \)
   \( = 3^4 \times 2^{-12} \)
   \( = \frac{81}{2^{12}} = \frac{81}{4096} \)

2.3. Упростите и найдите значение выражения

\( \left( -\frac{13}{15}a^4b^2 \right)^2 \cdot \left( \frac{15}{26}a^3b^4 \right)^3 \), если \( a = -1 \), \( b = \frac{6}{7} \)

\( \left( -\frac{13}{15}a^4b^2 \right)^2 = \left( -\frac{13}{15} \right)^2 (a^4)^2 (b^2)^2 = \frac{169}{225} a^8 b^4 \)
\( \left( \frac{15}{26}a^3b^4 \right)^3 = \left( \frac{15}{26} \right)^3 (a^3)^3 (b^4)^3 = \frac{3375}{17576} a^9 b^{12} \)
Теперь перемножим:
\( \frac{169}{225} a^8 b^4 \cdot \frac{3375}{17576} a^9 b^{12} = \frac{169 \times 3375}{225 \times 17576} a^{8+9} b^{4+12} \)
\( = \frac{169 \times (225 \times 15)}{225 \times (169 \times 104)} a^{17} b^{16} \)
\( = \frac{15}{104} a^{17} b^{16} \)
Теперь подставим \( a = -1 \) и \( b = \frac{6}{7} \):
\( \frac{15}{104} (-1)^{17} \left( \frac{6}{7} \right)^{16} \)
\( = \frac{15}{104} (-1) \left( \frac{6}{7} \right)^{16} \)
\( = -\frac{15}{104} \left( \frac{6}{7} \right)^{16} \)

Ответ: \( -\frac{15}{104} \left( \frac{6}{7} \right)^{16} \)

2.4. Разложите на множители:

1. а) \( 18a^2 - 2 = 2(9a^2 - 1) = 2(3a - 1)(3a + 1) \)


2. б) \( 2ax^3 - 16ay^3 = 2a(x^3 - 8y^3) = 2a(x - 2y)(x^2 + 2xy + 4y^2) \)


3. в) \( 4ay - 8a^2y + 4b^2y = 4y(a - 2a^2 + b^2) \)


4. г) \( 9m^2 - 6m - 10p - 25p^2 \)


5. д) \( 9x^2 + 9ax^2 - y^2 + ay^2 + 6axy \)
   \( = 9x^2(1+a) - y^2(1-a) + 6axy \)