I have been having problems with questions involving polynomials like asked in olympiads .
A few examples of these type of problems I'm posting here: (please don't give me solutions to these questions, cause I want to learn how to solve them not just remember their solutions)
Solve for real $x$: $\frac1{\lfloor x\rfloor} + \frac1{\lfloor 2x\rfloor}= \langle9x\rangle + \frac13\;,$ where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$ and $\langle x\rangle = x − \lfloor x\rfloor$.
Show that there are exactly $16$ pairs of integers $(x, y)$ such that $11x + 8y + 17 = xy$.
For a polynomial $f(x)$, let $f^{(n)}(x)$ denote the $n$-th–derivative for $n \ge 1$ and $f^{(0)}(x) = f(x)$. Is the following true or false ?
$f^{(n)}(a) = 0,\text{ for }n = 0, 1, \dots , k\quad \iff\quad (x−a)^{k+1}\text{ divides }f(x)$
Well, mainly the problems where you are given nothing but a few conditions that are to be applied on a polynomial of degree $n$.
Assuming that I have my basics clear in quadratic equations and polynomials Can anyone please refer me any book or any source from where I can learn to do these types of questions.