Find the factors of $(a-b)^{3}+(b-c)^{3}+(c-a)^{3}$.

If $x+\frac{1}{x}=3,$ then the value of $x^{6}+\frac{1}{x^{6}}$ is

Factors of $a^{2} + \frac{1}{4} + a$ will be

Let $f(x) = a_{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + \ldots + a_{n-1}x + a_{n}$,where $a_{0}, a_{1}, a_{2}, \ldots, a_{n}$ are constants. If $f(x)$ is divided by $ax - b$,then the remainder is:

If $x_{1} x_{2} x_{3} = 4(4 + x_{1} + x_{2} + x_{3})$,then what is the value of $\left[\frac{1}{(2 + x_{1})}\right] + \left[\frac{1}{(2 + x_{2})}\right] + \left[\frac{1}{(2 + x_{3})}\right]$?

When $4x^3 - ax^2 + bx - 4$ is divided by $x - 2$ and $x + 1$,the respective remainders are $20$ and $-13$. Find the values of $a$ and $b$.