# Leading term of monotonic polynomial shape proof with theorem?

Is there a theorem which concludes by the leading term the polynomial

$$p_n(x)=\sum_{i=0}^{n} a_i x^i$$

can predict information about the behavior/shape of the polynomial function? For a particular function with one real root (a zero crossing) and the rest being complex, is there a theorem that deduces that the graph behaves like a parabola for $$n$$ even and a curve with one crossing for $$p_n(x)=0$$ for $$n$$ odd. I have only seen practice examples around regarding leading coefficients with pictures as proofs, but no specific theorems to back them up.

• Sure, the leading term determines whether $\lim_{x\to\infty}p(x)$ and $\lim_{x\to-\infty}p(x)$ are positive or negative infinity (assuming degree 1 or more). Is this what you meant? – 79037662 Oct 4 '19 at 15:27

The leading term of a polynomial almost completely controls the behavior of the polynomial when $$x \gg 0$$ or $$x \ll 0$$ (much greater or much lesser).
Since a polynomial has only a finite number of coefficients, those coefficients are bounded. That is, there is some $$M$$ such that $$\left|\frac{a_i}{a_n}\right| < M$$ for all $$i$$. If we take, for example, $$|x| > 1000000nM$$, then each of the other terms of the polynomial is going to be at best $$1 / 1000000n$$ times the value of $$a_nx^n$$, and even their sum will be at most a millionth of the value of that leading term.
And as $$x$$ gets even larger in absolute value, the leading term becomes even more dominant. So no matter what polynomial $$p_n(x)$$ you have, if you back far enough away, it looks just like $$a_nx^n$$, except for tiny discrepancies near $$0$$.