Polynomial Functions

Now, you will learn how to identify a polynomial function and what makes them different from a power function. You will also be able to define the key characteristics of a polynomial function, such as the degree, leading coefficient, end behavior, intercepts, and turning points.

Identifying 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.

  • f(x) can be written as f(x)=6 x^{4}+4.
  • g(x) can be written as g(x)=-x^{3}+4 x.
  • h(x) cannot be written in this form and is therefore not a polynomial function.

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