Bolzano's theorem subset finding

Question

I have a continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ with $f(x)=x^5+x^3-x^2-2x+1$ and I want to prove that $f$ has at least one solution at $(-1,1)$. I am thinking of using Bolzano's theorem, but $f(-1)=0=f(1)$. I tried to find a subset of $(-1,1)$ and so far I have noticed that $f(0)=1>0$. I am trying now to find the other end of the subset that I want. Any ideas?

Answer 1

We have $$x^5+x^3-x^2-2x+1=(x-1)(x+1)(x^3+2x-1).$$ Since $x^3+2x-1$ increases, we see that this polynomial has unique real root.

Now, use the Bolzano's theorem.