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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. [latex-display]\left(w\right)=24+8w[/latex-display] We can combine this with the formula for the area A of a circle. [latex-display]\left(w\right)=\pi {r}^{2}[/latex-display] Composing these functions gives a formula for the area in terms of weeks. [latex-display]\begin{cases}\left(w\right)=\left(\left(\right)\right)\\ =\left(24+8w\right)\\ =\pi {\left(24+8w\right)}^{2}\end{cases}[/latex-display] Multiplying gives the formula. [latex-display]\left(w\right)=576\pi +384\pi w+64\pi {w}^{2}[/latex-display] 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 [latex-display]f\left(\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex-display] This is called the general form of a polynomial function. Each ${a}_{i}$ is a coefficient and can be any real number. Each product ${a}_{i}{x}^{i}$ is a term of a polynomial function.

Example 4: Identifying Polynomial Functions

Which of the following are polynomial functions? [latex-display]\begin{cases}f\left(x\right)=2{x}^{3}\cdot 3x+4\hfill \\g\left(x\right)=-x\left({x}^{2}-4\right)\hfill \\ h\left(x\right)=5\sqrt{x}+2\hfill \end{cases}[/latex-display]

Solution

The first two functions are examples of polynomial functions because they can be written in the form $f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}$, where the powers are non-negative integers and the coefficients are real numbers.
• $f\left(x\right)$ can be written as $f\left(x\right)=6{x}^{4}+4$.
• $g\left(x\right)$ can be written as $g\left(x\right)=-{x}^{3}+4x$.
• $h\left(x\right)$ cannot be written in this form and is therefore not a polynomial function.