Polynomials and Their Graphs

What counts as a polynomial

Look at three expressions: $3x^2 - 2x + 7$, $\dfrac{3}{x} + 1$, and $\tfrac{1}{2}x^3 + \sqrt{2}$. The first and the third are polynomials. The middle one is not, because $x$ sits in a denominator. What separates them is not whether a fraction or a square root appears anywhere in the expression: it is whether the variable $x$ is ever divided into, rooted, or placed in an exponent.

A term in the variable $x$ is an expression of the form $c\,x^k$, where the coefficient $c$ is a real number and the exponent $k$ is a whole number: $0, 1, 2, 3, \ldots$. A polynomial in $x$ is a finite sum of such terms. Writing $a_k$ for the coefficient that sits on $x^k$, the general shape is

and the two right-hand terms fit the pattern too, since $x^1 = x$ and $x^0 = 1$: the constant $a_0$ is the term $a_0 x^0$. You have been working with polynomials all along. A constant function is a polynomial, a linear function $mx + b$ is one, and a quadratic $ax^2 + bx + c$ is one; what is new is the license to keep climbing past the square.

This matches the description above exactly. Take any expression built from numbers and $x$ using only addition, subtraction, and multiplication, and multiply out every parenthesis. Multiplying powers adds their exponents ($x^2 \cdot x^3 = x^5$), so the exponents stay whole numbers, and a subtraction is just an addition with a minus sign built into the coefficient. Everything collapses into a sum of terms $c\,x^k$. So a polynomial is exactly the algebra you can do with $x$ without ever dividing by it or taking a root of it.

Two more expressions fail the same test:

• $\sqrt{x} + 5$ takes a root of the variable;
• $2^x$ puts the variable in the exponent (these exponential functions get their own chapter later in the course).

But the restriction falls only on $x$, never on the coefficients. The expression $\dfrac{x^2}{3} - \dfrac{x}{5} + \sqrt{2}$ is a perfectly good polynomial: the fractions and the square root are applied to numbers. Each term is still a real coefficient times a whole-number power of $x$.

Degree, leading coefficient, and standard form

Before naming a polynomial’s parts, tidy it: combine like terms, then write the terms in standard form, with the exponents descending from left to right. Once that is done, the degree is the highest power of $x$ that survives with a nonzero coefficient, and the leading term is the term carrying that power. The leading coefficient is that term’s coefficient, and the constant term is the term of degree zero, sign included.

The low degrees have names you know, and the next few are worth learning:

A second naming scheme counts terms instead of degree: a monomial has one term, a binomial two, a trinomial three. So $x^4 - 5x^2 + 4$ is a quartic trinomial, and $2x^5 + x$ is a quintic binomial. One special case sits outside the scheme: the zero polynomial $f(x) = 0$ has no nonzero term at all, so it gets no leading term and no degree.

Both tidying steps are load-bearing, and skipping either one produces wrong answers. Read the degree only after combining like terms: $(x^3 + 2x) - (x^3 - x)$ wears two cubes on its sleeve, but the cubes cancel and the difference is $3x$, a polynomial of degree $1$. And read the leading coefficient only after ordering the terms: the leading coefficient of $5x - 2x^4 + 3$ is not $5$ but $-2$, because standard form is $-2x^4 + 5x + 3$ and the $x^4$ term leads.

How does degree behave when polynomials combine? Addition is tame, multiplication is tamer, and both facts get used constantly for the rest of the course.

The multiplication rule means factored expressions surrender their degree and leading term without being expanded. The product $(1 - 2x)(x^2 + 3)^2$ has degree $1 + 2 + 2 = 5$, and its leading term is the product of the leading pieces, $(-2x)(x^2)(x^2) = -2x^5$. Keep that trick close; it powers the end-behavior readings later in this lesson.

A polynomial is a function

A polynomial is not just a string of symbols; it is a rule. Feed it an input and it returns an output, exactly the function machinery from Functions and Their Graphs. Powers, multiples, and sums accept every real number, so the domain of every polynomial function is all real numbers. There is no input to exclude, no division to go wrong, no root to turn negative.

Evaluating is substitution plus order of operations, and the one discipline worth drilling is to wrap negative inputs in parentheses before touching anything. Two evaluations are so cheap they are worth knowing by name. First, $f(0)$ kills every term carrying an $x$, so $f(0)$ is the constant term, and the graph’s $y$-intercept sits at that height. Second, every power of $1$ equals $1$, so $f(1)$ is the sum of the coefficients, a one-line check you will use more often than you expect.

Smooth and unbroken

Graph any polynomial and two qualities appear that no list of coefficients would lead you to expect. The curve is unbroken: you can draw it without lifting your pencil, with no gaps, jumps, or holes. That is because every real number is a legal input, and nudging the input a little nudges each term, and hence the output, only a little. And the curve is smooth: it bends, sometimes sharply, but it never creases into a corner the way $y = |x|$ does at the origin.

Both claims are proved properly in calculus; for now, trust them the way Graphs of Functions trusted smooth interpolation between plotted points. The contrast is what matters: no choice of coefficients gives a polynomial graph a sharp corner, and none gives it a break like the one $y = \dfrac{1}{x}$ has at $x = 0$.

Unbrokenness has one consequence this lesson will lean on twice. An unbroken curve cannot get from below the $x$-axis to above it without touching it on the way. If $f(a)$ is negative and $f(b)$ is positive, then somewhere between $a$ and $b$ the polynomial takes the value $0$. A sign change traps a zero.

End behavior: the leading term takes over

Now for the promise in the opening: one term controls both far ends of the graph. (This section is about polynomials with degree $n \ge 1$; a constant polynomial like $f(x) = 12$ has a horizontal graph with no far ends to speak of.) Watch it happen numerically for $f(x) = x^3 - 100x$, whose two terms pull in opposite directions.

Near the origin the $-100x$ term bullies the cube on both sides: at $x = \pm 1$ and $x = \pm 5$ the output has the opposite sign you would expect from a positive leading coefficient. But the cube grows like a cube while $100x$ grows only like a line, so by $x = \pm 20$ the cube has taken over. And at $x = \pm 100$ the correction is a one percent nick out of a million. The middle of a graph belongs to all the terms; the far ends belong to the leading term alone, one sign at each end here because $x^3$ itself is negative on the left and positive on the right.

Say the graph rises on an end when the outputs grow past any bound as you move along that end, and falls when they drop below any bound. Here is why the leading term decides both ends. Factor the highest power out of every term:

As $|x|$ grows, each fraction $\dfrac{a_k}{x^{n-k}}$ shrinks toward $0$, because dividing by an enormous number crushes anything finite. So far from the origin the bracket is close to $a_n$ alone, and $f(x)$ behaves like $a_n x^n$: the leading term, and nothing else.

Read off the sign of $a_n x^n$ and two conclusions drop out. When $n$ is even, $x^n > 0$ on both far ends, so both ends carry the sign of $a_n$. When $n$ is odd, $x^n$ keeps the sign of $x$ itself, so the far right carries the sign of $a_n$ and the far left carries the opposite sign.

Two switches, four silhouettes. The parity of the degree decides whether the two ends agree, and the sign of the leading coefficient decides which way the right end points:

The first two floors of the tower are the two positive-coefficient rows in miniature. A line with positive slope is the odd case ($n = 1$), falling on the left and rising on the right. Every upward parabola is the even case ($n = 2$), rising on both ends, exactly as chapter 4 read off the sign of $a$.

The odd rows hide a gift. An odd-degree polynomial falls without bound on one end and rises without bound on the other. So its unbroken graph starts below the $x$-axis and finishes above it, or the reverse. By the sign-change principle from the last section, it must touch the axis somewhere: every polynomial of odd degree has at least one real zero. No even-degree polynomial can make that promise; $x^2 + 1$ stays at height $1$ or above and never comes down to the axis.

Counting zeros and turning points

A real zero of $f$ is an input with $f(x) = 0$; on the graph it is an $x$-intercept, exactly as in Graphs of Functions. The degree does not tell you how many real zeros a polynomial has, but it does set a hard ceiling:

A polynomial of degree $n$ has at most $n$ real zeros.

You have already proved the first two cases in earlier chapters. A linear polynomial $mx + b$ with $m \ne 0$ has exactly one zero, $x = -\tfrac{b}{m}$, and no second one, because a nonzero slope never returns to a height it has left. A quadratic has at most two, as the quadratic formula and the discriminant made precise. The pattern behind the general bound is already visible in a factored cubic like $x(x - 2)(x + 2) = x^3 - 4x$: a product is zero exactly when a factor is. So the three linear factors hand over the three zeros $0$, $2$, and $-2$, and since degrees add, a degree-$n$ polynomial has room for at most $n$ linear factors. What is missing is the reverse direction, that every zero of a polynomial really does yield a linear factor. That missing link is the Factor Theorem, proved later in this chapter once long division is in hand. For now, use the ceiling as a counting tool, not a promise: $x^2 + 1$ has no real zeros at all, and $x^4$ has just the one, $x = 0$.

A turning point is a point where the graph switches between rising and falling. Such a point is the top of a local hill or the bottom of a local valley, like the vertex of a parabola. Turning points obey a twin ceiling:

The graph of a nonconstant polynomial of degree $n$ has at most $n - 1$ turning points.

(A constant polynomial, degree $0$, is the excluded case: its graph is a flat horizontal line with no turning points at all.) Check it against everything you know: a line ($n = 1$) never turns, and a parabola ($n = 2$) turns exactly once at its vertex. Meanwhile $y = x^3$ has no turning point at all, and $y = x^3 - 4x$ has two, one hill and one valley. The full proof of the ceiling belongs to calculus, where you will meet a companion polynomial of degree $n - 1$, the derivative, whose zeros include every turning point. The zeros ceiling applied to that companion gives the turning-point ceiling at once. What is within reach right now is the floor. Between two neighboring $x$-intercepts the graph leaves the axis and must come back to it, so it stops rising or stops falling somewhere in between. Each consecutive pair of zeros forces at least one turning point, so a polynomial with $k$ distinct real zeros carries at least $k - 1$ turning points.

The quartic in the figure is the same $x^4 - 5x^2 + 4$ whose end behavior you worked out in Graphs of Functions, now with its whole census on display. The census is four real zeros at $\pm 1$ and $\pm 2$ (it factors as $(x^2 - 1)(x^2 - 4)$), and three turning points, the maximum $4 - 1$ allows.

One more fact ties the ends to the turns, for any nonconstant polynomial (a constant graph has no ends that rise or fall, so it sits outside this count). Trace a polynomial’s graph from far left to far right: every turning point switches you from rising to falling or back, and nothing else does. When both ends point the same way, as in every even degree $n \ge 2$, you finish moving the opposite way you started, and only an odd number of switches can do that. So a graph of even degree $n \ge 2$ has an odd number of turning points, at least one. When the ends disagree, as in every odd degree, you finish moving the same way you started, which takes an even number of switches, including zero. That is exactly what $y = x^3$ does: it rises the whole way, with no turning point at all.

Common mistakes