What are Polynomial Functions with Degree Greater Than Two?

Polynomial functions with a degree greater than two are functions of the form f(x) = an x^n + a{n-1} x^{n-1} + ... + a_1 x + a_0, where 'n' is an integer greater than 2, and an, a{n-1}, ..., a_1, a_0 are constants with a_n ≠ 0. These functions produce curves when graphed, often oscillating above and below the x-axis, and are defined by their degree and coefficients.

Characteristics of Polynomial Functions with Degree > 2

• Degree: The highest power of x in the polynomial expression determines the degree. A degree greater than 2 means we're looking at cubic (degree 3), quartic (degree 4), quintic (degree 5) and higher order polynomials.
• Graphs: Unlike linear (degree 1) or quadratic (degree 2) functions that have straight lines or parabolas, these higher-degree polynomial functions exhibit more complex curves. They can have multiple turning points (local maxima and minima). The end behavior of the graph is determined by the leading coefficient and the degree.
• Zeros (Roots): The values of x where the function crosses or touches the x-axis are called zeros or roots. A polynomial of degree 'n' has at most 'n' real roots.
• Turning Points: Turning points are points where the graph changes direction (from increasing to decreasing or vice versa). A polynomial of degree 'n' can have at most n-1 turning points.
• Inflection Points: These are points where the concavity of the graph changes.

Examples

• Cubic Function (Degree 3): f(x) = x³ - 6x² + 11x - 6. The graph will have a general "S" shape.
• Quartic Function (Degree 4): f(x) = x⁴ - 4x³ + 2x² + 4x - 3. The graph can have a "W" or "M" shape or a variation of these.
• Quintic Function (Degree 5): f(x) = x⁵ - 5x³ + 4x. The graph will have a more complex wavy shape.

General Form

A general polynomial function of degree n can be expressed as:

f(x) = an x^n + a{n-1} x^{n-1} + ... + a_1 x + a_0

Where:

• n is a positive integer (the degree).
• a_n, a_{n-1}, ..., a_1, a_0 are constants (coefficients), and a_n ≠ 0.

Summary

Polynomial functions with a degree greater than two are defined by their degree, exhibit complex curves, can have multiple zeros and turning points, and are widely applicable in various fields such as engineering, physics, and economics for modeling complex relationships.