End Behavior of Polynomials

Definition A polynomial function is a function of the form

where $n$ is a nonnegative integer and the coefficients $a_0, a_1, \dots, a_n$ are real numbers. Its domain is all real numbers. When $a_n \ne 0$, the term $a_n x^n$ is the leading term, $a_n$ is the leading coefficient and $a_0$ is the constant term. Read these only after combining like terms and writing the powers in decreasing order, which is called standard form. The zero polynomial, with every coefficient $0$, has no leading term.

Definition The degree of a nonzero polynomial is the exponent $n$ of its leading term, the highest power of $x$ with a nonzero coefficient. A nonzero constant has degree $0$, and the zero polynomial is not assigned a degree. For nonzero polynomials $p$ and $q$, the degree of the product $pq$ is the sum of their degrees, and the leading coefficient of $pq$ is the product of their leading coefficients.

Definition The end behavior of a function $f$ describes what $f(x)$ does as $x \to \infty$, meaning as $x$ increases without bound, and as $x \to -\infty$, meaning as $x$ decreases without bound. The statement

means that for every number $K$ there is a number $M$ with $f(x) > K$ whenever $x > M$. The statement $f(x) \to -\infty$ as $x \to \infty$ means that for every $K$ there is an $M$ with $f(x) < K$ whenever $x > M$. The versions for $x \to -\infty$ use $x < M$ in place of $x > M$.

Definition Let $n$ be a positive integer and look at the power function $y = x^n$.

1. If $n$ is even, then $(-x)^n = x^n \ge 0$ for every $x$, and $x^n \to \infty$ both as $x \to \infty$ and as $x \to -\infty$.
2. If $n$ is odd, then $(-x)^n = -x^n$, and $x^n \to \infty$ as $x \to \infty$ while $x^n \to -\infty$ as $x \to -\infty$.

A polynomial has even degree or odd degree when its degree $n$ is even or odd. For degree $n \ge 1$, the leading coefficient test below shows that its tails follow the same pattern as $x^n$, turned upside down when the leading coefficient is negative.

Definition Let $p$ be a polynomial of degree $n \ge 1$ with leading coefficient $a_n$. Then $p$ has the same end behavior as its leading term $a_n x^n$.

The test says nothing about the middle of the graph, such as where the zeros or the turning points are.

Definition Informally, $\lim_{x \to \infty} f(x) = L$ means that $f(x)$ gets and stays as close to the number $L$ as you like once $x$ is large enough, and $\lim_{x \to \infty} f(x) = \infty$ is another way to write $f(x) \to \infty$ as $x \to \infty$. The same notation is used as $x \to -\infty$. For a polynomial of degree $n \ge 1$ and every $x \ne 0$,

Each fraction inside the bracket tends to $0$ as $x \to \infty$ and as $x \to -\infty$, so the bracket tends to $1$ and

In particular, once $\lvert x \rvert$ is large enough the bracket lies between $\frac{1}{2}$ and $\frac{3}{2}$. Then $p(x)$ has the same sign as $a_n x^n$ and $\lvert p(x) \rvert \ge \frac{1}{2} \lvert a_n \rvert \lvert x \rvert^n$, which is why the leading coefficient test holds. The precise definition of a finite limit at infinity comes in the lesson on horizontal asymptotes and again in the calculus lesson on limits at infinity.