So I want to prove that a polynomial $ P(x)=a_nx^n+a_{n−1}x^{n−1}+…..+a_1x+a_0 $ has a positive root. I'm given that $ a_n $ is positive and $ a_0 $ is negative. I want to know how to apply the intermediate value theorem EXACTLY.
Here is what the theorem says:
Let $f$ be a continuous real function on the interval $[a,b]$. If $f(a) \lt f(b)$ and if $c$ is a number such that $f(a) \lt c \lt f(b)$, then there exists a point $x$ in $(a,b)$ such that $f(x)=c$.
So I am assuming that polynomials are continuous. given that $P(0) \lt 0 \lt P(x)$ (for $x$ large enough). So there exists a point $c$ such that $f(c)=0$.
I have some gaps in my understanding of the argument..
Best Answer
"So there exists a point $c$ such that $f(c)=0$." That should be "There exists a point $y\in(0,x)$ such that $f(y)=c$." Afterall $c$ in the range value of $f$, and you want to find an $y$ for $c=0$.