Real solutions of the polynomial Let $a, b, c$ be distinct real numbers. Then find the number of real solutions of
$(x − a)^5 + (x − b)^3 + (x − c)$
I can't understand how there will be any solution. The polynomial is not equated with anything.

Assuming you equated it to $0$,
$$\lim_{x\to -\infty}f(x)=-\infty \text{ and }  \lim_{x\to \infty}f(x)=\infty$$ 
$$f'(x)=5(x-a)^4+3(x-b)^2+1>0\implies \text{Exactly one real root }$$
This is because the function is strictly increasing. It shall cross $x$-axis only one time.
Therefore, only one solution is possible. Perhaps a graph would help.
It is highly likely that this is the case.
