Polynomial End Behavior

The leading term is the whole story

As $x\to\pm\infty$, a polynomial behaves like its leading term $a_nx^n$ — every lower-degree term becomes negligible in comparison. End behavior is reported in limit notation, and there are exactly four cases:

• $n$ even, $a_n>0$: $\displaystyle\lim_{x\to-\infty}p(x)=\lim_{x\to+\infty}p(x)=+\infty$ (both ends up);
• $n$ even, $a_n<0$: both ends down ($-\infty$);
• $n$ odd, $a_n>0$: $\displaystyle\lim_{x\to-\infty}p(x)=-\infty$, $\displaystyle\lim_{x\to+\infty}p(x)=+\infty$;
• $n$ odd, $a_n<0$: the reverse — up on the left, down on the right.

Because the leading term dominates, ratios also collapse: $\displaystyle\lim_{x\to\infty}\dfrac{p(x)}{bx^n}=\dfrac{a_n}{b}$ whenever $p$ has degree $n$. The exam uses such limits to smuggle in the leading coefficient.

Assembling the full sketch

End behavior fixes the two tails; zeros with their multiplicities fix every meeting with the $x$-axis; the sign of $p$ between consecutive zeros then follows automatically. Those three ingredients determine the qualitative graph completely — no plotting required.