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Polynomial Functions

requiresQuadratic Functions & Parabolas ⚠Polynomial Arithmetic ⚠

⚗ Dr. Möbius, from the lab

You've been multiplying polynomial expressions. Fine, good for you. But now I want you to look at a polynomial as a function and tell me what the hell it's doing over all of $\mathbb{R}$ . What happens out at $x = \pm\infty$ ? How many times does it cross zero? Why does it bounce instead of cross at some roots, like a defeated specimen pressing its nose against the glass? The answers come straight from the structure you already built — degree, leading coefficient, and the Factor Theorem. Nothing new. Everything connected.

THE BIG IDEA

A polynomial function's global behavior is governed by its degree and leading coefficient, while its local behavior near each root is governed by that root's multiplicity.

End behavior: what happens at infinity

Pull back to the widest possible view. As $x \to \pm\infty$ , what dominates a polynomial? The leading term $a_n x^n$ — because the highest power of $x$ grows faster than every lower power and simply murders the rest. (This is an informal version of a limit argument; the rigorous version is in calculus, but the conclusion is available now and you should use it without hesitation.)

So for large $|x|$ , the polynomial $f(x) = a_n x^n + \text{lower terms}$ behaves like $a_n x^n$ . That's it. Two numbers control the far-left and far-right tails: the leading coefficient $a_n$ and the degree $n$ .

The four possible end behaviors:

Degree	Leading coeff	$x \to -\infty$	$x \to +\infty$
even	positive ( $a_n > 0$ )	$+\infty$	$+\infty$
even	negative ( $a_n < 0$ )	$-\infty$	$-\infty$
odd	positive ( $a_n > 0$ )	$-\infty$	$+\infty$
odd	negative ( $a_n < 0$ )	$+\infty$	$-\infty$

Mnemonic, if you need one: even-degree polynomials have "matching" tails (both up or both down); odd-degree have "opposite" tails. The sign of $a_n$ sets which way. This is not a rule to memorize — it follows from what $a_n x^n$ does when $n$ is even (always positive if $a_n > 0$ , regardless of the sign of $x$ ) versus odd ( $x^n$ has the sign of $x$ ).

Zeros, roots, and x-intercepts

A zero of $f$ is a value $c$ with $f(c) = 0$ . Equivalently, it's an $x$ -intercept of the graph. These are the same thing. The word "root" means the same thing. Three names for one concept — get used to it, mathematicians have absolutely no shame and have never once in history agreed on a single piece of terminology.

A degree- $n$ polynomial has at most $n$ real roots. Here's why: if $c_1, c_2, \ldots, c_k$ are distinct roots, then $(x - c_1)(x - c_2) \cdots (x - c_k)$ divides $f(x)$ (we'll prove this via the Factor Theorem below). That product is degree $k$ , and a degree- $k$ polynomial divides a degree- $n$ polynomial only if $k \le n$ . So $k \le n$ : at most $n$ roots.

The Factor Theorem

Theorem. $c$ is a zero of the polynomial $p(x)$ if and only if $(x - c)$ is a factor of $p(x)$ .

Proof sketch. Divide $p(x)$ by $(x - c)$ using polynomial long division. You get: $p(x) = (x - c) \cdot q(x) + r$ where $q(x)$ is the quotient (a polynomial of degree $\deg(p) - 1$ ) and $r$ is the remainder (a constant, since the divisor $(x-c)$ has degree 1). Now evaluate at $x = c$ : $p(c) = (c - c) \cdot q(c) + r = 0 + r = r.$ So the remainder $r = p(c)$ .

If $p(c) = 0$ : the remainder is $0$ , so $(x - c)$ divides $p(x)$ evenly. Factor found. If $(x - c)$ divides $p(x)$ : then $r = 0$ , so $p(c) = 0$ .

Both directions done. The Factor Theorem is just the Remainder Theorem applied to the case $r = 0$ .

This is your main tool for factoring higher-degree polynomials: test potential roots, and each confirmed root hands you a factor. It's free money. The proof cost us nothing. Use it shamelessly.

Multiplicity: crossing vs bouncing

When you write a polynomial in factored form, a factor might repeat: $f(x) = (x - 2)^3(x + 1)^2.$ The zero $x = 2$ has multiplicity 3; the zero $x = -1$ has multiplicity 2.

What does multiplicity do to the graph at a root?

Odd multiplicity: the graph crosses the $x$ -axis at that root. It passes through like a line through a point.
Even multiplicity: the graph bounces off the $x$ -axis (is tangent to it). It touches and turns back.

Why? Near $x = c$ with multiplicity $m$ , the function looks like $f(x) \approx A(x - c)^m$ for some nonzero constant $A$ . If $m$ is odd, $(x-c)^m$ changes sign as $x$ passes through $c$ , so $f$ changes sign — it crosses. If $m$ is even, $(x-c)^m$ is always non-negative (or non-positive), so $f$ doesn't change sign — it bounces. The graph touches the axis, thinks about crossing, and turns back like a coward. That's even multiplicity.

Drag the function grapher and watch this in action:

function grapher

x = 0.9(x-1)^2*(x+2) → 0.029(x-1)^3*(x+2) → -0.003

Sketching from factored form

Given $f(x) = a_n(x - c_1)^{m_1}(x - c_2)^{m_2} \cdots$ , you can sketch a rough graph:

End behavior: use $a_n$ and $n = \sum m_i$ .
Zeros: mark each $c_i$ on the $x$ -axis; cross (odd $m_i$ ) or bounce (even $m_i$ ).
Between zeros: the function maintains a sign (it can't change sign without passing through a zero). Check one test point in each interval.
Connect: draw a smooth curve consistent with all constraints.

This isn't a substitute for plotting software; it's a sanity-check machine. If your sketch says the function goes to $+\infty$ on the left but your end-behavior analysis says $-\infty$ , you've made an error somewhere. Catch it early — before it shows up on an exam and makes you feel like shit.

🔬 SPECIMENS (worked examples)

Worked example 1 — reading end behavior from the leading term▸

Determine the end behavior of $f(x) = -3x^5 + 7x^3 - 2x + 1$ .

The leading term is $-3x^5$ : degree $5$ (odd), leading coefficient $-3$ (negative).

Odd degree, negative leading coefficient: opposite tails, but flipped (negative sign reverses the standard odd-degree pattern).

Standard odd-degree with positive leading coefficient: $-\infty$ as $x \to -\infty$ , $+\infty$ as $x \to +\infty$ .

With negative leading coefficient, both tails flip: $x \to -\infty: \quad f(x) \to +\infty$ $x \to +\infty: \quad f(x) \to -\infty.$

Check with a specific value: $f(100) \approx -3(100)^5 < 0$ . Confirms $f(x) \to -\infty$ as $x \to +\infty$ .

Worked example 2 — applying the Factor Theorem to find a factor▸

Show that $x = 2$ is a zero of $p(x) = x^3 - 3x^2 + x + 2$ , then factor completely.

Step 1: verify the zero. $p(2) = 8 - 12 + 2 + 2 = 0. \checkmark$

Step 2: divide out the factor. By the Factor Theorem, $(x - 2)$ is a factor. Divide: $p(x) = x^3 - 3x^2 + x + 2 = (x - 2)(x^2 - x - 1).$

(Verification: $(x-2)(x^2 - x - 1) = x^3 - x^2 - x - 2x^2 + 2x + 2 = x^3 - 3x^2 + x + 2$ . Correct.)

Step 3: factor the quadratic. $x^2 - x - 1$ has discriminant $1 + 4 = 5 > 0$ . Roots: $x = \frac{1 \pm \sqrt{5}}{2}.$

These are irrational, so $x^2 - x - 1$ doesn't factor over $\mathbb{Q}$ . The complete factorization (over $\mathbb{R}$ ) is: $p(x) = (x - 2)\!\left(x - \frac{1+\sqrt{5}}{2}\right)\!\left(x - \frac{1-\sqrt{5}}{2}\right).$

Worked example 3 — sketching from multiplicity (don't you dare cross when you should bounce)▸

Sketch the rough behavior of $g(x) = x^2(x - 3)$ . Identify all zeros, their multiplicities, and the crossing/bouncing behavior at each.

Zeros. $g(x) = 0$ when $x = 0$ (multiplicity $2$ ) or $x = 3$ (multiplicity $1$ ).

End behavior. Leading term: $x^3$ . Degree $3$ , positive leading coefficient. Odd degree, positive: $f(x) \to -\infty$ as $x \to -\infty$ , $f(x) \to +\infty$ as $x \to +\infty$ .

Behavior at zeros.

$x = 0$ , multiplicity $2$ (even): graph bounces off the $x$ -axis.
$x = 3$ , multiplicity $1$ (odd): graph crosses the $x$ -axis.

Sign check. Test $x = 1$ : $g(1) = 1 \cdot (-2) = -2 < 0$ . So the graph is below the $x$ -axis on $(0, 3)$ .

Sketch. The curve comes from $-\infty$ on the left, rises, touches the $x$ -axis at $x = 0$ and bounces (stays below), crosses zero at $x = 3$ , then rises to $+\infty$ .

Key takeaway: $x = 0$ looks like the vertex of a parabola near the $x$ -axis — the graph doesn't pass through it but instead reverses direction. That's what even multiplicity does.

☠ KNOWN HAZARDS

Forgetting "at most" $n$ roots. A degree- $n$ polynomial has at most $n$ real roots, not exactly $n$ . A quartic might have only $2$ real roots (and $2$ complex ones). "At most" is doing real work in that sentence — ignore it and you'll be wrong.
Mixing up crossing vs bouncing. At a root of multiplicity 2, the graph is tangent to — not crossing — the $x$ -axis. At multiplicity 3, it crosses (and has an inflection-like bend there). Check the parity, not the magnitude, of the multiplicity. Even = bounces. Odd = crosses. Burn it in.
End behavior from the wrong term. The end behavior comes from the leading term, not from the constant term or any middle term. $f(x) = -x^4 + 1000x^3$ still goes to $-\infty$ at both ends because the $-x^4$ dominates for large $|x|$ . The $1000x^3$ is just noise at that scale. Ignore it.
Factor Theorem only works for LINEAR factors. It tells you $(x - c)$ is a factor when $p(c) = 0$ . It doesn't directly produce quadratic factors or higher — those require a separate investigation of the quotient $q(x)$ . The theorem is not magic; it's one tool in the kit, not the whole kit.

TL;DR

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End behavior is determined entirely by the leading term $a_n x^n$ : even degree means matching tails, odd means opposite; the sign of $a_n$ sets which direction.
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A degree- $n$ polynomial has at most $n$ real zeros.
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Factor Theorem: $(x - c)$ is a factor of $p(x)$ if and only if $p(c) = 0$ . Proof uses polynomial division and the remainder.
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Multiplicity of a root: odd multiplicity $\Rightarrow$ graph crosses; even multiplicity $\Rightarrow$ graph bounces (is tangent to the $x$ -axis).