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I rephrase my question how can prove or disprove that the roots of polynomial $P= x^{10}+ax^9+bx^8+cx^7+x+1$ can't be all real any way we choose $a,b, c\in \mathbb{R}$.

Erick Wong
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    what does it mean for a polynomial to be "splited"? – Chris Eagle Feb 20 '13 at 18:46
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    Have you tried using Descartes' Rule of Signs? – Pete L. Clark Feb 20 '13 at 18:50
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    Hard to prove, since it is false. As a consequence of the Fundamental Theorem of Algebra, any non-constant polynomial can be expressed as a product of linear and/or quadratic polynomials with real coefficients. – André Nicolas Feb 20 '13 at 18:51
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    @André: Sure. But then we come back to Chris's question: what does it mean for a polynomial to be "splited"? My guess was that it means that it splits into linear factors over the field, hence my comment above. Of course we really need to hear back from the OP... – Pete L. Clark Feb 20 '13 at 19:02

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As Pete commented, Descartes' rule of signs tells us that $P$ has at most four positive roots no matter how $a,b,c$ are chosen, and at most four negative roots no matter how $a,b,c$ are chosen. Therefore it must have at least two complex roots. (In fact it must have at least four complex roots, as it happens, since $a,b,c$ can't be chosen to simultaneously give four positive and four negative roots.) And an examination of a proof of the rule reveals that it does take multiple roots into account.

Greg Martin
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