Polynomial Functions: Structure, Zeros, and Graphs

Polynomial Functions

Definition of a Polynomial Function

A polynomial function is a function of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, where $n$ is a nonnegative integer and $a_n, a_{n-1}, \ldots, a_0$ are real numbers. The degree of the polynomial is $n$, and the leading coefficient is $a_n$.

• Degree: The highest power of $x$ in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest degree.

• Constant Term: The term $a_0$ with no variable.

Example: $f(x) = 2x^3 - 4x^2 + x - 5$ is a cubic polynomial (degree 3) with leading coefficient 2.

Graphs of Polynomial Functions – Smooth and Continuous

The graph of a polynomial function of degree 2 or higher is smooth (no sharp corners) and continuous (no breaks; can be drawn without lifting the pencil).

• Graphs contain only rounded curves.

• No gaps or jumps in the graph.

End Behavior of Polynomial Functions

The end behavior describes how the graph behaves as $x$ approaches positive or negative infinity. The sign of the leading coefficient and the degree of the polynomial determine this behavior.

• Even degree, positive leading coefficient: Both ends rise.

• Even degree, negative leading coefficient: Both ends fall.

• Odd degree, positive leading coefficient: Left end falls, right end rises.

• Odd degree, negative leading coefficient: Left end rises, right end falls.

Example: For $f(x) = x^4$, both ends rise since degree is even and leading coefficient is positive.

The Leading Coefficient Test

The Leading Coefficient Test helps predict the end behavior of a polynomial function:

• If $n$ is even and $a_n > 0$, $f(x) \to \infty$ as $x \to \pm\infty$.

• If $n$ is even and $a_n < 0$, $f(x) \to -\infty$ as $x \to \pm\infty$.

• If $n$ is odd and $a_n > 0$, $f(x) \to -\infty$ as $x \to -\infty$, $f(x) \to \infty$ as $x \to \infty$.

• If $n$ is odd and $a_n < 0$, $f(x) \to \infty$ as $x \to -\infty$, $f(x) \to -\infty$ as $x \to \infty$.

Zeros of Polynomial Functions

The zeros of a polynomial function are the values of $x$ for which $f(x) = 0$. These are also called roots or solutions of the polynomial equation. Each real zero corresponds to an x-intercept on the graph.

• To find zeros, set $f(x) = 0$ and solve for $x$.

• Factoring is a common method for finding zeros.

Example: For $f(x) = x^2 - 5x + 6$, set $x^2 - 5x + 6 = 0$ and factor to $(x-2)(x-3) = 0$, so zeros are $x = 2$ and $x = 3$.

Multiplicity and x-Intercepts

The multiplicity of a zero refers to how many times a particular root appears in the factorization of the polynomial.

• If $r$ is a zero of even multiplicity, the graph touches the x-axis at $r$ and turns around.

• If $r$ is a zero of odd multiplicity, the graph crosses the x-axis at $r$.

• Graphs flatten out near zeros with multiplicity greater than one.

Example: For $f(x) = (x-2)^2(x-3)$, $x=2$ has multiplicity 2 (touches and turns), $x=3$ has multiplicity 1 (crosses).

Turning Points of Polynomial Functions

A turning point is where the graph changes direction from increasing to decreasing or vice versa. For a polynomial of degree $n$, the graph has at most $n-1$ turning points.

• Degree 3: at most 2 turning points.

• Degree 4: at most 3 turning points.

Graphing a Polynomial Function: 3-Step Strategy

To graph a polynomial function, follow these steps:

1. Determine end behavior: Use the Leading Coefficient Test.

2. Find x-intercepts: Solve $f(x) = 0$ and determine multiplicities.

3. Find the y-intercept: Compute $f(0)$.

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

• Degree is 3 (odd), leading coefficient is 2 (positive): left end falls, right end rises.

• $x = -1$ (multiplicity 2): graph touches and turns at $x = -1$.

• $x = 3$ (multiplicity 1): graph crosses at $x = 3$.

• Find y-intercept: $f(0) = 2(0+1)^2(0-3) = 2 \cdot 1 \cdot (-3) = -6$.

Additional info: The graph above illustrates a cubic polynomial with two x-intercepts, one of which is a double root (touches and turns), and the other a single root (crosses). The end behavior matches the prediction for an odd-degree, positive leading coefficient polynomial.