# Method for proving polynomial inequalities

Let $x\in\mathbb{R}$. Prove that

$\text{(a) }x^{10}-x^7+x^4-x^2+1>0\\ \text{(b) }x^4-x^2-3x+5>0$

Possibly it can be proved in a few different ways, but I have first tried to prove it reducing to a sum of squares. After too many attempts and using a trial-and-error method, I got $$x^4\left({x^3-\frac12}\right)^2+\left({\frac12x^2-1}\right)^2+\frac12x^4>0$$ for $\text{(a)}$. My question is: is there any easier method to prove this for any polinomial which is always positive? Also, I am wondering if there is any other simplier method than reducing to sum of squares.

• If you've been taught differentiation, try and use that. Commented May 28, 2015 at 15:53
• Did you try taking derivatives and show they are positive? Commented May 28, 2015 at 15:53
• Of corse, I tried using derivatives, but I want method to prove any polynomial. If I have, for example, polynomial of degree $12$ and I take derivative, I again must prove it for polynomial of degree $11$.
– user164524
Commented May 28, 2015 at 15:54

We look only at $P(x)=x^{10}-x^7+x^4-x^2+1$. It is clear at a glance that $P(x)\gt 0$ if $|x|\ge 1$. Grouping as $(x^{10}-x^7)+(x^4-x^2)+1$ does it.

So we look at $|x|\lt 1$. Negative $x$ in this range are easy to deal with, so we concentrate on $0\lt x\lt 1$. Since $x^4-x^7\gt 0$, we have $P(x)\gt 1-x^2\gt 0$.

• Yes, it is correct. But, your solution doesn't answer my question. I am searching for the general method to prove this for any polynomial inequality. For example, how to prove that $x^{16}-3x^{15}+7x^{14}-4x^{13}-2x^{12}-7x^{11}+9x^{10}-8x^9+21x^8+7x^7+3x^6-17x^5+4x^3-x^2-x+1>0$ for all $x\in\mathbb{R}$?
– user164524
Commented May 28, 2015 at 16:06
• I was not attempting to answer the general question, except for in a highly indirect way advocating a "problem solving" approach in which we look at specific features of our object. From a logician's point of view, by a result of Tarski there is an algorithm for deciding whether a first order sentence in the usual language of ring theory is true in the reals. Commented May 28, 2015 at 16:16
• There is a fair bit of literature on efficient algorithms for deciding on positive definiteness, and semidefiniteness, for polynomials in $n$ variables. The old stuff in one variable was based on Sturm sequences. Commented May 28, 2015 at 16:34
• This is not exactlly what I want, but thank you anyway.
– user164524
Commented May 28, 2015 at 16:40
• You are welcome. Should really not accept yet, it diminishes the chance of getting a "real" solution. A Google search will get you algorithms. But maybe someone will save you the trouble. Commented May 28, 2015 at 16:46

It's not an easy method to do by hand, but one that always works is to use Sturm's theorem to find the number of real roots of the polynomial.

Just another way for $(a)$ is using the AM-GMs: $$\frac12x^{10}+\frac12x^4 \ge x^7, \quad \frac12x^4+\frac12 \ge x^2$$ $$\implies x^{10}-x^7+x^4-x^2+1 \ge \frac12+\frac12x^{10}>0$$

and similarly for $(b)$: $$\frac12x^4+\frac12 \ge x^2, \quad \frac12x^4+\frac32+\frac32+\frac32 \ge 2\times 3^{3/4}x> 3x$$

Not always applicable, of course.

Also you may want to know that for univariate polynomials theoretically you can always express as a sum of squares, and while tedious, somehow the end result is more satisfying...