2x^5+5x^4-13x^3-13x^2+5x+2=0

1. Вынесем общий множитель за скобки:

2x^5 + 5x^4 - 13x^3 - 13x^2 + 5x + 2 = 0; 2x^5 + 2 + 5x^4 + 5x - 13x^3 - 13x^2 = 0; 2 (x^5 + 1) + 5x (x^3 + 1) - 13x^2 (x + 1) = 0; 2 (x + 1) (x^4 - x^3 + x^2 - x + 1) + 5x (x + 1) (x^2 - x + 1) - 13x^2 (x + 1) = 0; (x + 1) (2x^4 - 2x^3 + 2x^2 - 2x + 2 + 5x^3 - 5x^2 + 5x - 13x^2) = 0; (x + 1) (2x^4 + 3x^3 - 16x^2 + 3x + 2) = 0.

a)

x + 1 = 0; x = - 1;

b)

2x^4 + 3x^3 - 16x^2 + 3x + 2 = 0; 2x^4 + 2 + 3x^3 + 3x - 16x^2 = 0; 2x^2 + 2/x^2 + 3x + 3/x - 16 = 0; 2 (x + 1/x) ^2 - 4 + 3 (x + 1/x) - 16 = 0; 2 (x + 1/x) ^2 + 3 (x + 1/x) - 20 = 0.

2. Введем новую переменную:

x + 1/x = t; 2t^2 + 3t - 20 = 0; D = 3^2 + 4 * 2 * 20 = 9 + 160 = 169 = 13^2; t = (-3 ± 13) / 4; t1 = (-3 - 13) / 4 = - 16/4 = - 4; t2 = (-3 + 13) / 4 = 10/4 = 5/2.

3. Вернемся к переменной x:

x + 1/x = t; x^2 + 1 = tx; x^2 - tx + 1 = 0;

1) t = - 4;

x^2 + 4x + 1 = 0; D/4 = 2^2 - 1 = 3; x = - 2 ± √3;

2) t = 5/2;

x^2 - 5/2 * x + 1 = 0; 2x^2 - 5x + 2 = 0; D = 5^2 - 4 * 2 * 2 = 25 - 16 = 9 = 3^2; x = (5 ± 3) / 4; x1 = (5 - 3) / 4 = 2/4 = 1/2; x2 = (5 + 3) / 4 = 8/4 = 2.

Ответ: - 1; - 2 ± √3; 1/2; 2.