Let $P \in \mathbb R[x]$ be a degree-$n$ polynomial with real coefficients such that $P(a) \neq 0$, where $a$ is real. If $P'(a) = P ''(a) = 0$ then prove that $P$ cannot have all roots real.

Can someone suggest a possible solution using Rolle's Theorem. All I could gather was that $P'(x) = 0$ has a repeated root by Rolle's Theorem. But I am stuck after this

  • If $P(x)$ can have $n$ real roots, then by Rolle's theorem you would get $P^{(n)}$ has some real root, which is impossible, cause $P^{(n)}$ is a nonzero constant. – xbh Aug 16 at 7:25
  • @Arthur Sorry about the ambiguity. If $P$ has $n$ real roots and $P'(a)= P''(a)=0$ where $P(a)\neq 0$, then $P'$ would have $n$ roots [count multiplicity], then $P^{(n)}$ would have root. Am I correct now? – xbh Aug 16 at 7:31
  • @xbh If you can prove that $P'$ has $n$ real roots, then you don't need to go to $P^{(n)}$ to prove contradiction; $P'$ has degree $n-1$ and therefore cannot have $n$ roots. I think the main point of this exercise, however, is proving that $P'$ has at least $n$ roots in this case. You've just pointed it out as though it's a triviality and then spent a few lines proving the part which is (relatively) trivial. – Arthur Aug 16 at 7:39
  • @Arthur Thanks. I do not know what the OP has learned, so I wrote a few more lines. If s/he knew about the fact, then my lines are truly not necessary. – xbh Aug 16 at 7:44
up vote 25 down vote accepted

Assume that $P$ has degree $n$ and let $x_1,x_2,\dots,x _n$ be all its roots (repetitions are allowed). Then $P(x)=c\prod_{k=1}^n (x-x_k)$, and if $x$ is not a root of $P$ we have that $$\frac{P'(x)}{P(x)}=\sum_{k=1}^n \frac{1}{x-x_k}.$$ After taking the derivative we obtain $$\frac{P''(x)P(x)-(P'(x))^2}{(P(x))^2}=-\sum_{k=1}^n \frac{1}{(x-x_k)^2}.$$ Finally by letting $x=a$ (which is not a root) we get a contradiction: $$0=\frac{P''(a)P(a)-(P'(a))^2}{(P(a))^2}=-\sum_{k=1}^n \frac{1}{(a-x_k)^2}<0$$ where the right-hand side is negative because $a, x_1,x_2,\dots,x _n$ are all real.

Sketch of proof: Assume all roots of $P$ are real, and let $x_1\leq x_2\leq \ldots\leq x_n$ be the $n$ roots (with repetition if $P$ has repeated roots). What does Rolle's theorem say about the roots of $P'$? How many roots does $P'$ have (counted with multiplicity)? Can $P'$ have a repeated root which is not one of the $x_i$?

I guess $P$ is not constant, otherwise the statement is false: all the roots of the constant $a$ polynomial are real, as everything holds for the elements of the emptyset.

Translate the polynomial by $a$, i.e., $Q(x):= P(x-a)$. Then the conditions can be rephrased to $Q$ equivalently as follows: $Q(0)\neq 0$, $Q'(0)=Q''(0)=0$. In other words, $Q(x)= a_nx^n+ \cdots +a_3x^3 + a_2x^2+a_1x+a_0$, where $a_0\neq 0$ and $a_1=a_2=0$.

Write the Viete formulas for the roots $x_i$:

$\prod x_i= (-1)^na_0\neq 0, \prod x_i \cdot \sum 1/x_i=0$, and $\prod x_i \cdot \sum\limits_{i\neq j} 1/(x_ix_j)=0$.

Put $y_i=1/x_i$ (possible, as $0$ is not a root, as the constant term is nonzero), then after simplification, you obtain $\sum y_i= \sum\limits_{i\neq j} y_iy_j=0$, but then $\sum y_i^2= 0$. So if these are real numbers, then all the $y_i$ are zero, a contradiction.

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