Functions
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Polynomial Functions

Degree, roots, and end behavior — all visible on one graph

Roller coaster curves, economic models, signal processing — when a straight line or parabola is not enough, polynomials step in. A polynomial is a sum of terms, each being a constant times a power of x: f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0. The highest power is called the degree, and its coefficient an is the leading coefficient.

The degree and leading coefficient control the end behavior — what happens as x goes to positive or negative infinity. An odd-degree polynomial rises on one end and falls on the other; an even-degree polynomial rises (or falls) on both ends.

In this lesson you'll explore polynomials of degree 2, 3, and 4 with sliders, discover how the number of roots connects to the degree, and see how the leading coefficient flips the curve.

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FAQ

What is a polynomial function?
A polynomial function is a function of the form f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, where each ai is a constant and n is a non-negative integer called the degree. Examples: x² + 3x − 1 (degree 2), 2x³ − x (degree 3).
What is the degree of a polynomial?
The degree is the highest power of x that appears with a non-zero coefficient. It determines the maximum number of roots and the general shape: degree 2 makes a parabola, degree 3 makes an S-curve, degree 4 can have a W-shape.
What is end behavior?
End behavior describes what happens to f(x) as x → ∞ and x → −∞. For even degree with positive leading coefficient, both ends go UP. For odd degree with positive leading coefficient, the left end goes DOWN and the right end goes UP. A negative leading coefficient flips everything.
How many roots can a polynomial have?
A polynomial of degree n has at most n real roots (x-intercepts). It can have fewer if some roots are complex (non-real). For example, x² + 1 has degree 2 but zero real roots because x² + 1 > 0 for all real x.

Related Topics

Quadratic EquationsPower FunctionsFactoring & Roots