Question

What are two reasons why an odd degree function is guaranteed at least one real root?

Answer 1

Let us suppose that
\[
f(x)= a x^3 + b x^2 + c x + d
\]
is the function. Suppose that a >0 for a moment.
If x approaches negative infinity, f(x) will do the same. So there will be an p for which f(p) <0
If x approaches positive infinity, f(x) will do the same. So there will be an q for which f(q) >0
By the intermediate value theorem, the should be a point z between p and q so that f(z)=0

Answer 2

You can do similar argument if a<0

Answer 3

Thank you

Answer 4

YW