1. What a Polynomial Function Means
A polynomial function is a function that can be written using powers of x with real coefficients. These powers are whole numbers (0, 1, 2, 3, ...). The function is built from terms like c, ax, bx², cx³, and so on.
2. General Form of a Polynomial Function
A polynomial function in one variable x looks like this:
2.1. Expression
f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0
\text{where } a_n, a_{n-1}, ..., a_0 \text{ are real constants and } n \in \mathbb{N} \cup \{0\}
3. Degree of a Polynomial Function
The degree of a polynomial is the highest power of x that appears with a non-zero coefficient. The degree tells a lot about the shape and behaviour of the function.
3.1. Examples
- Degree 0 → constant polynomial
- Degree 1 → linear polynomial
- Degree 2 → quadratic polynomial
- Degree 3 → cubic polynomial
- Degree n → nth degree polynomial
4. Examples of Polynomial Functions
These are common polynomial functions:
4.1. Example 1 — Linear
f(x) = 3x + 7
4.2. Example 2 — Quadratic
f(x) = x^2 - 4x + 5
4.3. Example 3 — Cubic
f(x) = x^3 - 2x + 1
4.4. Example 4 — Higher Degree
f(x) = 5x^4 - x^2 + 6
5. What Makes a Function Not a Polynomial
A function is not polynomial if:
5.1. Conditions
- x appears in the denominator (e.g., \( 1/x \))
- x appears inside a square root (e.g., \( \sqrt{x} \))
- x appears in a power that is not a whole number (e.g., x1/2, x-1)
- There are trigonometric, exponential, or logarithmic functions involved
6. Basic Properties of Polynomial Functions
Polynomial functions have nice and predictable behaviour.
6.1. 1. Defined for All Real x
No restrictions such as division by zero or square roots of negatives.
6.2. 2. Smooth and Continuous
No breaks, holes, or sharp corners in the graph.
6.3. 3. Degree Controls the Shape
Even-degree polynomials open upward or downward; odd-degree polynomials extend in opposite directions.
7. Why Polynomial Functions Matter
Polynomial functions form the foundation of algebra and calculus. Their predictable behaviour makes them useful for graphing, modelling, approximations, and solving many types of equations.