$x^3 - x = 0$
$x(x^2 - 1) = 0$
$x(x - 1)(x + 1) = 0$
$x_1 = 0$;
$x_2 - 1 = 0$
$x_2 = 1$;
$x_3 + 1 = 0$
$x_3 = -1$.

$x^4 + x^2 = 0$
$x^2(x^2 + 1) = 0$
$x_1^2 + 1 = 0$
$x_1^2 ≠ -1$, так квадрат числа не может быть числом отрицательным;
$x_2^2 = 0$
$x_2 = 0$.

$x^4 - 8x^3 = 0$
$x^3(x - 8) = 0$
$x_2^2 = 0$
$x_1^3 = 0$
$x_1 = 0$;
$x_2 - 8 = 0$
$x_2 = 8$.

$49x^3 + 14x^2 + x = 0$
$x(49x^2 + 14x + 1) = 0$
$x(7x + 1)^2 = 0$
$x_1 = 0$;
$(7x_2 + 1)^2 = 0$
$7x_2 + 1 = 0$
$7x_2 = -1$
$x_2 = -\frac{1}{7}$.

$x^3 + x^2 - x - 1 = 0$
$(x^3 + x^2) - (x + 1) = 0$
$x(x + 1) - (x + 1) = 0$
$(x + 1)(x - 1) = 0$
$x_1 + 1 = 0$
$x_1 = -1$;
$x_2 - 1 = 0$
$x_2 = 1$.

$x^3 - 4x^2 - 25x + 100 = 0$
$(x^3 - 4x^2) - (25x - 100) = 0$
$x^2(x - 4) - 25(x - 4) = 0$
$(x - 4)(x^2 - 25) = 0$
$(x - 4)(x - 5)(x + 5) = 0$
$x_1 - 4 = 0$
$x_1 = 4$;
$x_2 - 5 = 0$
$x_2 = 5$;
$x_3 + 5 = 0$
$x_3 = -5$.