Show that the polynomial $x^{8}-x^{7}+x^{2}-x+15$ has no real root.

If $x<0$ every term is positive, hence the polynomial is $>0$.

If $x>1$ , then $x^2-x>0$ , $x^8 - x^7 >0$ , $15>0$ hence the function is $>0$.

If $0<x<1$, then $1-x>0$ , $x^2 - x^6>0$, $14+x^8>0$ hence the function is $>0$

Hence for every real $x$ the function is $>0$ , hence no real root.

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Polynomials

Problem Solving

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