Polynomial Zeros, Signs, and Graph Shape

Definition A polynomial in $x$ is a function of the form

where $n$ is a nonnegative integer and the coefficients $a_0, \ldots, a_n$ are real numbers. If $a_n \ne 0$, the degree of $p$ is $n$ and $a_n$ is the leading coefficient. Every polynomial is defined for all real $x$, and its graph is one unbroken curve with no holes, jumps or vertical asymptotes.

Definition A zero, or root, of a polynomial $p$ is a number $c$ with $p(c) = 0$. A real zero $c$ gives the point $(c, 0)$ where the graph meets the $x$-axis, an $x$-intercept. A polynomial of degree $n \ge 1$ has at most $n$ real zeros. Some polynomials have no real zeros at all.

Definition A polynomial $d$ is a factor of $p$ when $p = d \cdot q$ for some polynomial $q$. The factor theorem says that for a real number $c$,

Factoring a polynomial into linear factors and quadratic factors with no real zeros lists all of its real zeros.

Definition A real zero $c$ of a nonzero polynomial $p$ has multiplicity $m$ when

for a polynomial $q$ with $q(c) \ne 0$. In other words, $m$ is the number of copies of the factor $x - c$ in the complete factorization. A zero of multiplicity $1$ is called simple.

Definition Let $c$ be a real zero of $p$ with multiplicity $m$, written as $p(x) = (x - c)^m q(x)$ with $q(c) \ne 0$. A polynomial graph has no breaks, so $q$, which is not $0$ at $c$, stays nonzero and keeps one sign on some open interval around $c$ that contains no other zero of $p$. On that interval the sign of $p$ is the sign of $(x - c)^m$ times that fixed sign.

• If $m$ is odd, the factor $(x - c)^m$ changes sign at $c$, so $p$ changes sign and the graph crosses the axis.
• If $m$ is even, the factor $(x - c)^m$ is positive on both sides of $c$, so $p$ keeps its sign and the graph touches the axis and turns back.

A zero of odd multiplicity $m \ge 3$ still crosses, but the graph flattens against the axis as it passes through.

Definition A sign chart of a polynomial is a number line marked with its real zeros, together with the sign of the polynomial on each of the open intervals into which those zeros cut the line, including the two unbounded intervals at the ends. A polynomial keeps one sign on each of these intervals. The reason is that its graph has no breaks, so if it were positive at one point and negative at another point of the same interval, it would have to equal $0$ somewhere between them. This is the intermediate value theorem, which calculus proves for every continuous function. Since the interval contains no zero, the sign cannot change. You find the sign either from the signs of the factors or from one test value per interval.

Definition The leading term of a polynomial of degree $n \ge 1$ is $a_n x^n$. For $x \ne 0$ the polynomial can be written as

and every fraction in the parentheses tends to $0$ as $\lvert x \rvert$ grows. So for large $\lvert x \rvert$ the polynomial is close to its leading term in relative size, and has the same sign. In factored form the leading term is the product of the leading terms of the factors.

Definition The end behavior of a polynomial describes $p(x)$ as $x \to \infty$ and as $x \to -\infty$. For degree $n \ge 1$ it is the end behavior of the leading term $a_n x^n$.

• If $n$ is even, both ends go the same way, up to $\infty$ when $a_n > 0$ and down to $-\infty$ when $a_n < 0$.
• If $n$ is odd, the ends go opposite ways. When $a_n > 0$ the graph falls on the left and rises on the right, and when $a_n < 0$ it rises on the left and falls on the right.