Question

use the intermediate value theorem to prove that
p(x)=X^7+x^6+x^5+x^4+x^3+x^2+x+1
has at least one zero

Answer 1

Try plugging in values like \(x=0\) and \(x=-2\). You won't find a positive root to this, because \(p(\text{any positive})=\text{positive}\).

At \(x=0\), you have \(p(0)=0+0+0+0+0+0+0+1=1\),
at \(x=-1\), you have \(p(-1)=-1+1-1+1-1+1-1+1=0\),
and so on.

To apply the IVT, you need only show that \(p(x)\) attains a negative number.