Expand the following algebraic expressions: 1. (x+2)(y-6) 2. (x+2)(y-6) 3. (4-a)(5-b) 4. (α-2)(3-α) 5. (2x-1)(3+4x) 6. (3b-7)(1+7b) 7. (5+3x)(2x-7) 8. (2α+3)(5+6α) 9. (1+8b)(6b-1) 10. (-2-α)(b+7) 11. (-5-n)(2x+3) 12. (-3-t)(4t-9) 13. (2α+7)(-3-α) 14. (α+b)(4α-5b) 15. (2x-4y)(x-y) 16. (-6-v)(2v-5) 17. (4u+v)(3v-u) 18. (-v-8u)(5v-u) 19. (2b-7)(-b+6) 20. (5t+3)(-t-7) 21. (-z-6)(4z-3)

Решение:

Для раскрытия скобок будем использовать дистрибутивное свойство умножения (каждый член первой скобки умножается на каждый член второй скобки).

1. \( (x+2)(y-6) = x(y-6) + 2(y-6) = xy - 6x + 2y - 12 \)
2. \( (x+2)(y-6) = xy - 6x + 2y - 12 \)
3. \( (4-a)(5-b) = 4(5-b) - a(5-b) = 20 - 4b - 5a + ab \)
4. \( (\alpha-2)(3-\alpha) = \alpha(3-\alpha) - 2(3-\alpha) = 3\alpha - \alpha^2 - 6 + 2\alpha = -\alpha^2 + 5\alpha - 6 \)
5. \( (2x-1)(3+4x) = 2x(3+4x) - 1(3+4x) = 6x + 8x^2 - 3 - 4x = 8x^2 + 2x - 3 \)
6. \( (3b-7)(1+7b) = 3b(1+7b) - 7(1+7b) = 3b + 21b^2 - 7 - 49b = 21b^2 - 46b - 7 \)
7. \( (5+3x)(2x-7) = 5(2x-7) + 3x(2x-7) = 10x - 35 + 6x^2 - 21x = 6x^2 - 11x - 35 \)
8. \( (2\alpha+3)(5+6\alpha) = 2\alpha(5+6\alpha) + 3(5+6\alpha) = 10\alpha + 12\alpha^2 + 15 + 18\alpha = 12\alpha^2 + 28\alpha + 15 \)
9. \( (1+8b)(6b-1) = 1(6b-1) + 8b(6b-1) = 6b - 1 + 48b^2 - 8b = 48b^2 - 2b - 1 \)
10. \( (-2-\alpha)(b+7) = -2(b+7) - \alpha(b+7) = -2b - 14 - \alpha b - 7\alpha \)
11. \( (-5-n)(2x+3) = -5(2x+3) - n(2x+3) = -10x - 15 - 2nx - 3n \)
12. \( (-3-t)(4t-9) = -3(4t-9) - t(4t-9) = -12t + 27 - 4t^2 + 9t = -4t^2 - 3t + 27 \)
13. \( (2\alpha+7)(-3-\alpha) = 2\alpha(-3-\alpha) + 7(-3-\alpha) = -6\alpha - 2\alpha^2 - 21 - 7\alpha = -2\alpha^2 - 13\alpha - 21 \)
14. \( (\alpha+b)(4\alpha-5b) = \alpha(4\alpha-5b) + b(4\alpha-5b) = 4\alpha^2 - 5\alpha b + 4\alpha b - 5b^2 = 4\alpha^2 - \alpha b - 5b^2 \)
15. \( (2x-4y)(x-y) = 2x(x-y) - 4y(x-y) = 2x^2 - 2xy - 4xy + 4y^2 = 2x^2 - 6xy + 4y^2 \)
16. \( (-6-v)(2v-5) = -6(2v-5) - v(2v-5) = -12v + 30 - 2v^2 + 5v = -2v^2 - 7v + 30 \)
17. \( (4u+v)(3v-u) = 4u(3v-u) + v(3v-u) = 12uv - 4u^2 + 3v^2 - uv = -4u^2 + 11uv + 3v^2 \)
18. \( (-v-8u)(5v-u) = -v(5v-u) - 8u(5v-u) = -5v^2 + uv - 40uv + 8u^2 = 8u^2 - 39uv - 5v^2 \)
19. \( (2b-7)(-b+6) = 2b(-b+6) - 7(-b+6) = -2b^2 + 12b + 7b - 42 = -2b^2 + 19b - 42 \)
20. \( (5t+3)(-t-7) = 5t(-t-7) + 3(-t-7) = -5t^2 - 35t - 3t - 21 = -5t^2 - 38t - 21 \)
21. \( (-z-6)(4z-3) = -z(4z-3) - 6(4z-3) = -4z^2 + 3z - 24z + 18 = -4z^2 - 21z + 18 \)

Ответ: 1. \( xy - 6x + 2y - 12 \)
2. \( xy - 6x + 2y - 12 \)
3. \( 20 - 4b - 5a + ab \)
4. \( -\alpha^2 + 5\alpha - 6 \)
5. \( 8x^2 + 2x - 3 \)
6. \( 21b^2 - 46b - 7 \)
7. \( 6x^2 - 11x - 35 \)
8. \( 12\alpha^2 + 28\alpha + 15 \)
9. \( 48b^2 - 2b - 1 \)
10. \( -2b - 14 - \alpha b - 7\alpha \)
11. \( -10x - 15 - 2nx - 3n \)
12. \( -4t^2 - 3t + 27 \)
13. \( -2\alpha^2 - 13\alpha - 21 \)
14. \( 4\alpha^2 - \alpha b - 5b^2 \)
15. \( 2x^2 - 6xy + 4y^2 \)
16. \( -2v^2 - 7v + 30 \)
17. \( -4u^2 + 11uv + 3v^2 \)
18. \( 8u^2 - 39uv - 5v^2 \)
19. \( -2b^2 + 19b - 42 \)
20. \( -5t^2 - 38t - 21 \)
21. \( -4z^2 - 21z + 18 \)