If $x = 2 + 2^{2/3} + 2^{1/3},$ then $x^3 - 6x^2 + 6x = $

The number of solutions of the equations $x+y+z=1$,$x^2+y^2+z^2=1$,and $x^3+y^3+z^3=1$ is:

The number of positive integers $x$ satisfying the equation $\frac{1}{x} + \frac{1}{x+1} + \frac{1}{x+2} = \frac{13}{12}$ is:

The condition that ${x^3} - 3px + 2q$ may be divisible by a factor of the form ${x^2} + 2ax + {a^2}$ is

To remove the second term of the equation $x^4-8x^3+x^2-x+3=0$,diminish the roots of the equation by

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?
$I$. It has at least one negative root.
$II$. It has at least one positive root.
$III$. Both its roots are real.