Question

Prove that x^3+1/x^2+1=5 has no roots in [0,2)

Answer 1

Consider the polynomial g=(x−1)(1+x+x2+x3+⋯+xn)

As f∣g,g all the roots of f are roots of g.

This means I have to prove the statement:

If c is a multiple root of f⟹c is multiple a root of g.

Equivalently:

If c is not a multiple root of g⟹c is not a multiple root of f.

I'll try to prove this last statement:

Suppose ∃c such that c is a multiple root of g, then

g(c)=0⟹g′(c)=0

g=(x−1)(1+x+⋯+xn)⟹g′=nxn

g′(c)=ncn=0⟹c=0

But

g(0)≠0

And we've found the contradiction: ∄c:g(c)=g′(c)=0. Then f has no multiple roots.

Is this correct?

Also: Assuming this is correct, is it possible to prove ∑nk=0xnn! has no multiple roots in a similar manner?