Identify polynomial functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius r of the spill depends on the number of weeks w that have passed. This relationship is linear.
(w)=24+8w
We can combine this with the formula for the area A of a circle.
(w)=πr2
Composing these functions gives a formula for the area in terms of weeks.
{(w)=(())=(24+8w)=π(24+8w)2
Multiplying gives the formula.
(w)=576π+384πw+64πw2
This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

A General Note: Polynomial Functions

Let n be a non-negative integer. A polynomial function is a function that can be written in the form
f()=anxn+⋯+a2x2+a1x+a0
This is called the general form of a polynomial function. Each ai is a coefficient and can be any real number. Each product aixi is a term of a polynomial function.

Example 4: Identifying Polynomial Functions

Which of the following are polynomial functions?
{f(x)=2x3⋅3x+4g(x)=−x(x2−4)h(x)=5x+2

Solution

The first two functions are examples of polynomial functions because they can be written in the form f(x)=anxn+⋯+a2x2+a1x+a0, where the powers are non-negative integers and the coefficients are real numbers.

  • f(x)
    can be written as f(x)=6x4+4.
  • g(x)
    can be written as g(x)=−x3+4x.
  • h(x)
    cannot be written in this form and is therefore not a polynomial function.