Existence of a polynomial in between two other polynomials

Question

Suppose we have some interval $I \subset \mathbb{R}$, $I = [a,b]$ for some $a,b$, $a<b$. Suppose there also exists two polynomials P and Q, where for all x in I, P < Q. Can we say that there exists another polynomial R, s.t. for all $x \in I$, $P<R<Q$.

If this is trivial, under what conditions does this become none-trivial?

I have thought a lot but I believe my current maths skills cannot solve this. Could anyone help me?

Answer 1

Your condition on $R$ seems to be missing, but judging by the title of your question, perhaps $R=(P+Q)/2$ would do the job ? It is always strictly between $P$ and $Q$, on $I$.

Answer 2

I think it should read: " s.t. for all $x \in I, P < R < Q$."

If yes, try $R:= \frac{P+Q}{2}.$

Answer 3

since $P<Q$ There will be at least one $R$ that satisfies $P<R<Q$ if $P$ and $Q$ are both positive. proof here: https://www.math.wustl.edu/~freiwald/310ratbetweenreals.pdf