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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.

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  • $\begingroup$ If you've been taught differentiation, try and use that. $\endgroup$
    – George Moore
    Commented May 28, 2015 at 15:53
  • $\begingroup$ Did you try taking derivatives and show they are positive? $\endgroup$
    – SalmonKiller
    Commented May 28, 2015 at 15:53
  • $\begingroup$ 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 $11$. $\endgroup$
    – user164524
    Commented May 28, 2015 at 15:54

3 Answers 3

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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 $P(x)\gt 1-x^2\gt 0$.

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  • $\begingroup$ 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 $x\in\mathbb{R}$? $\endgroup$
    – user164524
    Commented May 28, 2015 at 16:06
  • $\begingroup$ 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. $\endgroup$
    – André Nicolas
    Commented May 28, 2015 at 16:16
  • $\begingroup$ 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. $\endgroup$
    – André Nicolas
    Commented May 28, 2015 at 16:34
  • $\begingroup$ This is not exactlly what I want, but thank you anyway. $\endgroup$
    – user164524
    Commented May 28, 2015 at 16:40
  • $\begingroup$ 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. $\endgroup$
    – André Nicolas
    Commented May 28, 2015 at 16:46
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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.

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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...

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