If p(x) is a polynomial of odd degree, show that the equation p(x)=0 has at least one solution. how would I do this?
Odd degree functions come in two cases, both depending on their highest degree terms.
We know that
Everything is reversed from example one. The limit as
Hopefully this helps!
Doesn't quite fit in a sentence, check below.
Since polynomial functions are continuous everywhere, it would be enough to find two points "at" which the polynomial has a different sign (one at which it's positive, one at which it's negative, and let's call them
Our polynomial of odd degree has the general form
Let's take the limits at positive and negative infinity of
The second line is explained because
From the above limits we can see that there will be a value for
*Now if you were to consider the case where