Originally Posted by

**Rugevit**
Well done!

There are many solutions to the problem. One possible solution is to use the summing of geometric series. Geometric series are still being taught in schools?

So, х^12 - x^9 + x^6 - x^3 + 1 is sum of a geometric series with the coefficient -x^3. Therefore, х^12 - x^9 + x^6 - x^3 + 1 = (x^15+1)/(x^3+1) > 0 for all real x >0

Alternatively, one may do some curly algebra re-writing х^12 - x^9 + x^6 - x^3 + 1 as (x^6-x^3 +1)^2 + (x^3)(x^3-1)^2 > 0 for all real x > 0

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Let's make the problem a bit more difficult proving that х^12 - x^9 + x^6 - x^3 + 1 >0 for all real x.