How many real roots does x⁵ − 5x + a have?

Critical points. $P_a'(x)=5(x^2-1)(x^2+1)$ vanishes only at $x=\pm1$. Hence $P_a$ is increasing on $(-\infty,-1]$, decreasing on $[-1,1]$ and increasing on $[1,+\infty)$.

Values. $P_a(-1)=4+a$ (local maximum) and $P_a(1)=a-4$ (local minimum). Moreover $P_a(x)\to-\infty$ as $x\to-\infty$ and $P_a(x)\to+\infty$ as $x\to+\infty$.

Counting. On each of the three monotone pieces there is at most one root, and by the intermediate value theorem there is one exactly when the endpoint values (or limits) have opposite signs or one is zero.
- $|a|<4$: $P_a(-1)>0>P_a(1)$. One root in each piece: three roots.
- $a=4$: $P_a(-1)=8>0$ and $P_a(1)=0$. The root $x=1$ is a double root and there is one more root on $(-\infty,-1)$: two distinct roots.
- $a=-4$: symmetric: $P_a(-1)=0$ is a double root and there is one root on $(1,\infty)$: two distinct roots.
- $a>4$: $P_a(1)>0$, so $P_a>0$ on $[-1,\infty)$. One root, on $(-\infty,-1)$: one root.
- $a<-4$: symmetric, one root on $(1,\infty)$.

Lesson. To count the real roots of a polynomial depending on a parameter, study the critical values: roots appear or disappear exactly when a critical value crosses $0$.