Polynomials

Ask the tutor about Polynomials Open the graphing workbench

The power function $y = a x^{n}$ is a single term. Add lower powers of $x$ , and the graph can wiggle in the middle. Far to the left and far to the right, who still decides whether the arms rise or fall?

The leading term still wins far away

Keep the degree odd and freeze a concrete cubic: $y = a x^{3} - x$ , with the leading coefficient $a$ held off zero. Drag $a$ . Far out, the curve still follows $a x^{3}$ : the lower powers become negligible compared with the leading term.

Drag the leading coefficient a. The green curve is y = a x³ − x. Far left and far right follow a x³. When a stays off zero the graph meets the x-axis at least once; the marks sit on the real zeros of this cubic.

When $a > 0$ the left arm comes from below and the right arm leaves upward; when $a < 0$ those ends swap. An odd-degree graph with nonzero leading coefficient therefore takes opposite signs at $+ \infty$ and $- \infty$ . A continuous curve that does that must meet the x-axis at least once. The blue marks on this board are the real zeros of this particular cubic.

Name the whole finite sum

An $n$ th-degree real polynomial is a function of the form

$y = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0},$

where $n$ is a nonnegative integer $0, 1, 2, \dots$ , every coefficient $a_{k}$ is real, and the leading coefficient satisfies $a_{n} \neq = 0$ .

At every real $x$ , this $y = f (x)$ is continuous and infinitely differentiable. Its first derivative is again a polynomial, of degree $n - 1$ when $n \geq 1$ :

$f^{'} (x) = n a_{n} x^{n - 1} + (n - 1) a_{n - 1} x^{n - 2} + \dots + a_{1} .$

Limits at infinity follow $a_{n} x^{n}$

For $n \geq 1$ , as $x \to \pm \infty$ the graph of $f$ behaves like the power $y = a_{n} x^{n}$ . Explicitly,

$x \to + \infty lim f (x) = {+ \infty, - \infty, a_{n} > 0, a_{n} < 0,$

and

$x \to - \infty lim f (x) = {+ \infty, - \infty, a_{n} > 0 and n even, or a_{n} < 0 and n odd, a_{n} > 0 and n odd, or a_{n} < 0 and n even .$

So the sign of $a_{n}$ and the parity of $n$ fix both ends. That is why an odd $n$ forces at least one real zero: the two limits disagree in sign.

Even degree picks a global extreme

Switch to even degree. Drag the leading coefficient on $y = a x^{4} - 2 a x^{2}$ . When $a > 0$ the arms open upward; the lower blue marks are global minima — there is a point $a_{*}$ with $f (a_{*}) \leq f (x)$ for every real $x$ . Flip $a$ below zero and the same shape turns over, so the extreme becomes a global maximum.

Drag the leading coefficient a. The green curve is y = a x⁴ − 2 a x². When a > 0 the arms open upward and the lower marks are global minima; flip a below zero and that extreme becomes a global maximum.

In general: if $n$ is even and $a_{n} > 0$ , then $y = f (x)$ has a global minimum; if $n$ is even and $a_{n} < 0$ , it has a global maximum. Odd degree cannot lock a global min or max on the whole line, because one arm already escapes to $+ \infty$ and the other to $- \infty$ .

Local extrema alternate; inflections stay few

Degree also caps how much the graph can wiggle. For $n \geq 2$ , $y = f (x)$ has at most $n - 1$ local extrema, and local minima and local maxima appear alternately. For $n \geq 3$ , the graph has at most $n - 2$ inflection points.

Drag c. The green curve is y = x³ − c x. The orange marks are a local max and a local min that swap sides of the origin; the blue mark at the origin is the inflection, and the dashed line is the tangent there (slope −c).

Drag $c$ on $y = x^{3} - c x$ . Here $n = 3$ , so at most two local extrema and at most one inflection. The orange marks are a local max and a local min on opposite sides of the origin; the blue mark at the origin is the inflection, where the dashed tangent of slope $- c$ shows the change of concavity. Raise $c$ and the extrema move apart; the single inflection stays put. That matches the bound: two extrema, one inflection — and they alternate in the local-max / local-min sense along the curve.

What this page put in place

A polynomial is a finite sum with a nonzero leading term. It is smooth on the whole line; far away it matches $a_{n} x^{n}$ ; odd degree meets the axis at least once; even degree carries a global min or max according to the sign of $a_{n}$ ; and the counts of local extrema and inflection points are bounded by the degree. The next natural object is a quotient of such sums — rational functions.