positive (negative) intervals for polynomials with complex roots

Question

I have a question, which could be high-school level, sorry about that in advance, but could not find a source on the net. Here is my question:

Suppose that we have fifth order polynomials with one real root $x=a,$ and other four roots are complex.
$f(x) = \sum_{i=0}^{5}A_ix^i$ such that let also assume that $A_5>0$ and $f(a+1)>0$, and $f(a-1)<0$.
Can we say that only critical point is $0$, and this function is positive [negative] for interval $(a,\infty) [(-\infty,a)]?$

In other words, when I analyze where the function is positive or negative, should I worry about complex roots, or just looking at real roots? My intuition says that since these complex roots would never cut $x-$axis, it should not change the sign of the function. Any idea (some formal references would be much appreciated as well)? Thanks in advance.

Answer 1

You are correct in disregarding the complex roots in determining the sign of your function.

It suffices to only consider the real roots and check the intervals between them for the sign of the function.

However the intervals of increase or decrease are found by the sign of the first derivative of the function not the location of zeros of the function.

For example the function $$f(x)=x^2+4$$ Has complex roots and it's interval of increase is determined by the the sign of $$f'(x)=2x$$

Another example $$f(x)=x^5-3x^4+x-5$$

The intervals of increase or decrease are found by the sign of $$f'(x)=5x^4-12x^3+1$$