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.

$r(w) \equiv 24+8 w$

We can combine this with the formula for the area $A$ of a circle.

$A(r)=\pi r^{2}$

Composing these functions gives a formula for the area in terms of weeks.

$\begin{aligned} A(w) &=A(r(w)) \\ &=A(24+8 w) \\ &=\pi(24+8 w)^{2} \end{aligned}$

Multiplying gives the formula.

$A(w)=576 \pi+384 \pi w+64 \pi w^{2}$

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.

POLYNOMIAL FUNCTIONS

Let $n$ be a non-negative integer. A polynomial function is a function that can be written in the form

$f(x)=a_{n} x^{n}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}$

This is called the general form of a polynomial function. Each $a_{i}$ is a coefficient and can be any real number, but $a_{n} \neq$. Each expression $a_{i} x^{i}$ is a term of a polynomial function.

EXAMPLE 4

Identifying Polynomial Functions

Which of the following are polynomial functions?

$\begin{aligned} f(x) &=2 x^{3} \cdot 3 x+4 \\ g(x) &=-x\left(x^{2}-4\right) \\ h(x) &=5 \sqrt{x+2} \end{aligned}$

Solution

The first two functions are examples of polynomial functions because they can be written in the form $f(x)=a_{n} x^{n}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}$, where the powers are non-negative integers and the coefficients are real numbers.

Source: Rice University, https://openstax.org/books/college-algebra/pages/5-2-power-functions-and-polynomial-functions
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