Polynomial Functions

Definition

A polynomial function is a function of the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a non-negative integer and the coefficients $a_i$ are real numbers. The degree is $n$, the highest power of $x$.

• How to read: “The polynomial function f of x is equal to the coefficient a n times x raised to the nth power, plus lower degree terms, down to the constant term a zero; and the degree of the polynomial is n.”
• Meaning: Finite sum of power terms—degree $n$ controls end behavior and max turning points ($n-1$).

Why It Matters

Polynomials model systems where many factors compete. The “Leading Term” is the destiny of the system.

Core Concepts

• Smoothness and Continuity: Polynomial graphs are “well-behaved”—they have no gaps, holes, or sharp corners.
• Zeros and Intercepts:
  • $x$-intercepts occur at the real zeros of the function ($f(x)=0$).
  • How to read: “The x-intercepts occur at the values of x where the function f of x is equal to zero.”
  • Meaning: Roots of the polynomial—graph crosses or touches the x-axis.

Connected Concepts

• Zeros of Polynomial Functions
• Polynomial Graphs
• Polynomials: Basic Operations