Решите уравнение: x5 + 2x4 - 3x3 - 6x2 + 2x + 4 = 0

x^5 + 2x^4 - 3x^3 - 6x^2 + 2x + 4 = 0;

(x^5 + 2x^4 - 3x^3) + ( - 6x^2 + 2x + 4) = 0;

x^3 (x^2 + 2x - 3) - 2 (3x^2 - x - 2) = 0.

Разложим на множители выражения (x^2 + 2x - 3) и (3x^2 - x - 2).

x^2 + 2x - 3 = 0;

D = b^2 - 4ac;

D = 4 + 4 * 3 = 4 + 12 = 16; √D = 4;

x = (-b ± √D) / (2a);

x1 = (-2 + 4) / 2 = 1;

x2 = (-2 - 4) / 2 = - 3;

x^2 + 2x - 3 = (x - 1) (x + 3).

3x^2 - x - 2 = 0;

D = b^2 - 4ac;

D = 1 + 4 * 3 * 2 = 25; √D = 5;

x = (-b ± √D) / (2a);

x1 = (1 + 5) / 6 = 1;

x2 = (1 - 5) / 6 = - 4/6 = - 2/3;

3x^2 - x - 2 = 3 (x + 2/3) (x - 1) = (3x + 2) (x - 1).

x^3 (x - 1) (x + 3) - 2 (3x + 2) (x - 1) = 0;

(x - 1) (x^3 (x + 3) - 2 (3x + 2)) = 0;

x - 1 = 0;

x1 = 1 - первый корень;

x^3 (x + 3) - 2 (3x + 2) = 0;

x^4 + 3x^3 - 6x - 4 = 0;

(x^4 - 4) + (3x^3 - 6x) = 0;

(x^2 - 2) (x^2 + 2) + 3x (x^2 - 2) = 0;

(x^2 - 2) (x^2 + 2 + 3x) = 0;

x^2 - 2 = 0;

x^2 = 2;

x2 = √2 - второй корень;

x3 = - √2 - третий корень;

x^2 + 2 + 3x = 0;

D = b^2 - 4ac;

D = 9 - 4 * 2 = 1; √D = 1;

x = (-b ± √D) / (2a);

x4 = (-3 + 1) / 2 = - 1 - четвертый корень;

x5 = (-3 - 1) / 2 = - 2 - пятый корень.

Ответ. 1; - 1; - 2; √2; - √2.