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.

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    $\begingroup$ 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? $\endgroup$ – 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 $0$.


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