Polynomial Functions

You have already met polynomials. In Algebra 1, you added them, subtracted them, multiplied them, and factored them. You spent a lot of time with quadratics - those degree-2 polynomials with their parabola graphs. Now it is time to go bigger. What happens when the highest power is 3? Or 4? Or 7? The curves get more interesting, with more hills and valleys, more places where they cross the x-axis. Understanding how these higher-degree polynomials behave gives you the tools to analyze complex relationships that appear everywhere - from the trajectory of a spacecraft to the shape of a roller coaster.

Think about a roller coaster for a moment. The track goes up, comes down, goes up again, maybe dips below the starting point, then climbs once more. That kind of path - with multiple peaks and valleys - cannot be described by a simple line or even a parabola. You need a polynomial of higher degree. And here is the remarkable thing: once you understand a few key features, you can sketch the general shape of any polynomial just by looking at its equation.

Core Concepts

Definition and Key Terminology

A polynomial function is a function of the form:

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$$

where $n$ is a non-negative integer and the coefficients $a_n, a_{n-1}, \ldots, a_0$ are real numbers with $a_n \neq 0$.

That looks complicated, but it is just saying: add up terms where $x$ is raised to various whole number powers, each multiplied by some constant. The key requirements are:

• Only whole number exponents (no $x^{1/2}$ or $x^{-1}$)
• A finite number of terms
• The leading coefficient (the one with the highest power) cannot be zero

The degree of the polynomial is the highest power of $x$. This single number tells you more about the polynomial’s behavior than almost anything else.

The leading coefficient is the coefficient of the highest-degree term. Together with the degree, it determines what happens to the graph as $x$ gets very large or very small.

For example, in $f(x) = -2x^5 + 3x^3 - x + 7$:

• The degree is 5 (the highest power)
• The leading coefficient is $-2$ (the coefficient of $x^5$)
• There are four terms

End Behavior: What Happens at the Extremes

One of the most useful things to understand about a polynomial is its end behavior - what happens to $f(x)$ as $x$ goes toward positive infinity ($x \to +\infty$) and negative infinity ($x \to -\infty$).

Here is the beautiful simplicity: end behavior depends only on two things - the degree and the sign of the leading coefficient. All those other terms become insignificant when $x$ gets very large.

Think about $f(x) = 2x^4 - 1000x^3 + 5000x^2$. When $x = 10$, the $x^4$ term gives $20{,}000$, the $x^3$ term gives $-1{,}000{,}000$, and the $x^2$ term gives $500{,}000$. The $x^3$ term dominates! But when $x = 1000$, the $x^4$ term gives $2 \times 10^{12}$, while the $x^3$ term gives only $-10^{12}$. As $x$ grows, the highest power always wins eventually.

The End Behavior Rules:

Degree	Leading Coefficient	Left End ($x \to -\infty$)	Right End ($x \to +\infty$)	Shape Description
Even	Positive ($+$)	$f(x) \to +\infty$	$f(x) \to +\infty$	Both ends up
Even	Negative ($-$)	$f(x) \to -\infty$	$f(x) \to -\infty$	Both ends down
Odd	Positive ($+$)	$f(x) \to -\infty$	$f(x) \to +\infty$	Falls left, rises right
Odd	Negative ($-$)	$f(x) \to +\infty$	$f(x) \to -\infty$	Rises left, falls right

A helpful way to remember this:

• Even degree: Both ends go the same direction (like a parabola opening up or down)
• Odd degree: The ends go opposite directions (like a line with positive or negative slope, but curvier)
• Positive leading coefficient: The right end goes up
• Negative leading coefficient: The right end goes down

Zeros, Roots, and X-Intercepts

The zeros of a polynomial function $f(x)$ are the values of $x$ that make $f(x) = 0$. These are also called roots or x-intercepts (because they are where the graph crosses or touches the x-axis).

Finding zeros is one of the central problems in working with polynomials. When a polynomial is in factored form, the zeros jump right out at you:

$$f(x) = (x - 2)(x + 3)(x - 5)$$

The zeros are $x = 2$, $x = -3$, and $x = 5$ - just set each factor equal to zero and solve.

A polynomial of degree $n$ has at most $n$ real zeros. It might have fewer (some zeros could be complex numbers), but it cannot have more. A degree-4 polynomial might have 4 real zeros, or 2, or 0, but never 5.

Multiplicity: Touch or Cross?

When a factor appears more than once, we say the corresponding zero has multiplicity greater than 1. This multiplicity determines how the graph interacts with the x-axis at that zero.

For $f(x) = (x - 2)^3(x + 1)^2$:

• The zero $x = 2$ has multiplicity 3 (the factor $(x-2)$ appears 3 times)
• The zero $x = -1$ has multiplicity 2 (the factor $(x+1)$ appears 2 times)

The Multiplicity Rule:

Multiplicity	Graph Behavior at Zero	Description
1 (odd)	Crosses the x-axis	The graph goes straight through
2 (even)	Touches and turns	The graph bounces off like a parabola
3 (odd)	Crosses with flattening	The graph passes through but flattens out
4 (even)	Touches and turns (flatter)	Like multiplicity 2, but flatter

The pattern: odd multiplicity means crossing, even multiplicity means touching (bouncing).

Higher multiplicities make the curve flatter at the zero. A zero with multiplicity 3 creates a subtle “S-curve” as the graph passes through, while a zero with multiplicity 1 crosses more steeply.

Turning Points

A turning point is where the graph changes from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum). These are the peaks and valleys of the polynomial.

A polynomial of degree $n$ has at most $n - 1$ turning points.

• A degree-2 polynomial (quadratic) has at most 1 turning point (the vertex)
• A degree-3 polynomial (cubic) has at most 2 turning points
• A degree-4 polynomial (quartic) has at most 3 turning points

This makes sense when you think about it: each turning point represents a change in direction, and a polynomial of degree $n$ can only “wiggle” so many times.

Note that the polynomial might have fewer than $n - 1$ turning points. For example, $f(x) = x^3$ has degree 3, so it could have up to 2 turning points, but it actually has 0 - it just keeps rising.

The Factor Theorem

The Factor Theorem provides a powerful connection between factors and zeros:

Factor Theorem: $(x - c)$ is a factor of polynomial $f(x)$ if and only if $f(c) = 0$.

This works in both directions:

• If you know $(x - 3)$ is a factor of $f(x)$, then $f(3) = 0$
• If you know $f(3) = 0$, then $(x - 3)$ is a factor of $f(x)$

This is incredibly useful. To test whether $(x - 2)$ is a factor of $x^3 - 4x^2 + x + 6$, just calculate $f(2)$. If you get zero, it is a factor. If you get anything else, it is not.

The Remainder Theorem

The Remainder Theorem is closely related:

Remainder Theorem: When polynomial $f(x)$ is divided by $(x - c)$, the remainder equals $f(c)$.

If you divide $f(x)$ by $(x - 3)$ and want to know the remainder without doing long division, just calculate $f(3)$. Whatever you get is the remainder.

Notice that the Factor Theorem is actually a special case of the Remainder Theorem: if the remainder is 0, then $(x - c)$ divides evenly into $f(x)$, making it a factor.

Graphing Polynomials

To sketch a polynomial graph, gather these key features:

1. End behavior: Use the degree and leading coefficient
2. Y-intercept: Set $x = 0$ and calculate $f(0)$
3. Zeros: Find where $f(x) = 0$ (factor if possible)
4. Multiplicity of zeros: Determine if the graph crosses or touches at each zero
5. Turning points: Know the maximum number ($n - 1$)

Then connect the dots with a smooth curve that respects the end behavior and multiplicity rules.

Writing Polynomials from Zeros

If you know the zeros of a polynomial, you can write the polynomial in factored form. For each zero $c$, include a factor $(x - c)$.

If the zeros are $x = 2$, $x = -1$, and $x = 5$, then the polynomial has factors $(x - 2)$, $(x + 1)$, and $(x - 5)$:

$$f(x) = a(x - 2)(x + 1)(x - 5)$$

The constant $a$ accounts for vertical stretching - the zeros alone do not determine this. If you are also given a point on the graph, you can solve for $a$.

When a zero has multiplicity greater than 1, the corresponding factor gets raised to that power. If $x = 3$ is a zero with multiplicity 2, the factor is $(x - 3)^2$.

Notation and Terminology

Term	Meaning	Example
Degree	Highest power of variable	$x^4 + 3x^2 - 1$ has degree 4
Leading coefficient	Coefficient of highest-degree term	In $2x^3 - x$, it is 2
Zero/Root	Value where $f(x) = 0$	If $f(x) = x^2 - 4$, zeros are $\pm 2$
Multiplicity	How many times a factor repeats	$(x - 2)^3$ has zero $x = 2$ with multiplicity 3
End behavior	What happens as $x \to \pm\infty$	Degree 3 with positive leading coeff: falls left, rises right
Turning point	Where graph changes direction	Polynomial of degree $n$ has at most $n - 1$
Factor Theorem	$(x-c)$ is a factor iff $f(c) = 0$
Remainder Theorem	Remainder when dividing by $(x-c)$ equals $f(c)$

Examples

Key Properties and Rules

End Behavior Summary

The end behavior of $f(x) = a_n x^n + \ldots$ depends only on:

1. The degree $n$ (even or odd)
2. The sign of $a_n$ (positive or negative)

Remember: even degree = both ends same direction; odd degree = ends opposite directions.

The Factor-Zero Connection

• Factor Theorem: $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$
• Remainder Theorem: When $f(x)$ is divided by $(x - c)$, the remainder is $f(c)$

Multiplicity and Graph Behavior

Multiplicity	Behavior at x-axis
1 (or any odd)	Graph crosses
2 (or any even)	Graph touches and bounces

Higher multiplicity = flatter curve at that zero.

Maximum Number of Features

For a polynomial of degree $n$:

• At most $n$ real zeros
• At most $n - 1$ turning points

Writing Polynomials from Zeros

If zeros are $c_1, c_2, \ldots, c_k$ with multiplicities $m_1, m_2, \ldots, m_k$:

$$f(x) = a(x - c_1)^{m_1}(x - c_2)^{m_2} \cdots (x - c_k)^{m_k}$$

where $a$ is determined by an additional point or condition.

Common Mistakes to Avoid

• Confusing degree and number of terms - The degree is the highest exponent, not the number of terms. $x^5 + 1$ has degree 5 but only 2 terms.
• Forgetting the sign in factors - A zero at $x = 3$ gives factor $(x - 3)$, not $(x + 3)$
• Mixing up touch vs. cross - Even multiplicity touches; odd multiplicity crosses
• Assuming all zeros are real - A degree-4 polynomial might have only 2 real zeros (the other 2 are complex)

Real-World Applications

Roller Coaster Design

When engineers design roller coasters, they use polynomial functions to model the track. The hills and valleys you experience are turning points of the polynomial. A degree-6 polynomial can have up to 5 turning points, allowing for a ride with multiple peaks and drops. The designer chooses zeros (where the track crosses a certain height) and adjusts multiplicity to control whether the track smoothly touches that height and turns back, or crosses through it.

Economic Models

Economists use polynomial functions to model cost, revenue, and profit relationships. For example, a company’s profit function might be:

$$P(x) = -0.001x^3 + 0.5x^2 - 50x - 1000$$

where $x$ is the number of units produced. The zeros tell you break-even points (where profit is zero), and the turning points reveal production levels that maximize or minimize profit. The cubic term captures how efficiency changes at different scales.

Population Dynamics

Biologists model population changes with polynomial functions when simple exponential models are not accurate. A population might follow:

$$P(t) = -0.02t^4 + 0.5t^3 - 2t^2 + 10t + 100$$

The zeros (where they exist for positive $t$) might represent extinction points, and the turning points show when population growth rates change from increasing to decreasing or vice versa.

Curve Fitting in Data Analysis

When scientists collect data that does not follow a simple pattern, they often fit a polynomial to the data points. A polynomial of degree $n$ can pass through exactly $n + 1$ points. This technique, called polynomial interpolation, is used in everything from climate modeling to financial forecasting. The challenge is choosing the right degree - too low and you miss important features; too high and you get unrealistic wiggles.

Computer Graphics and Bezier Curves

The smooth curves you see in fonts, car designs, and animation are often defined using polynomial curves called Bezier curves. When you use the pen tool in graphic design software, you are actually manipulating the control points of cubic (degree-3) polynomial curves. The mathematical properties of polynomials - smoothness, predictable behavior, easy computation - make them ideal for computer graphics.

Self-Test Problems

Summary

• A polynomial function has terms with variables raised to whole number powers. The degree is the highest power, and the leading coefficient is the coefficient of that term.

• End behavior depends only on degree and leading coefficient. Even degree: both ends go the same way. Odd degree: ends go opposite ways. Positive leading coefficient: right end rises. Negative: right end falls.

• The zeros (or roots) of a polynomial are where $f(x) = 0$. A degree-$n$ polynomial has at most $n$ real zeros.

• Multiplicity tells you how many times a factor repeats. Odd multiplicity means the graph crosses the x-axis at that zero; even multiplicity means the graph touches and bounces back.

• A polynomial of degree $n$ has at most $n - 1$ turning points (peaks and valleys).

• The Factor Theorem says $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$. This gives you a quick test for factors.

• The Remainder Theorem says that when you divide $f(x)$ by $(x - c)$, the remainder equals $f(c)$.

• To graph a polynomial, find the end behavior, zeros (and their multiplicities), and y-intercept. Connect with a smooth curve that respects touch vs. cross behavior at each zero.

• To write a polynomial from zeros, use the factor form: for each zero $c$ with multiplicity $m$, include the factor $(x - c)^m$.

You now have the tools to analyze polynomial functions of any degree. Whether you are modeling real-world phenomena, solving equations, or sketching graphs, these concepts - degree, leading coefficient, zeros, multiplicity, end behavior - are your fundamental toolkit. When you see a polynomial, you know how to read its story.