Number of roots for polynomial

Question

For the polynomial $p(x)= 8x^{10}-7x^{3}+x-1$ consider the following statements (which may be true or false)

(i) It has a root between [0, 1].

(ii) It has a root between [0, -1].

(iii) It has no roots outside (-1, 1).

Which of the above statements are true?

Only (i).

Only (i) and (ii).

Only (i) and (iii).

Only (ii) and (iii).

All of (i), (ii) and (iii).

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It is easy to see that there is a root between [0,1] and [0,-1], But how to figure out if It has no roots outside (-1, 1) or it has ?

Answer 1

If $x>1$, then $8x^{10}>7x^3$, so $p(x)>0$.

If $x<-1$, then $\underbrace{8x^{10}}_{>0}+\underbrace{x}_{<0}>0$ and $\underbrace{-7x^3}_{>0}-1>0$, so again $p(x)>0$.