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 [latex]{a}_{i}[/latex] is a coefficient and can be any real number. Each product [latex]{a}_{i}{x}^{i}[/latex] 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 [latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex], where the powers are non-negative integers and the coefficients are real numbers.- [latex]f\left(x\right)[/latex] can be written as [latex]f\left(x\right)=6{x}^{4}+4[/latex].
- [latex]g\left(x\right)[/latex] can be written as [latex]g\left(x\right)=-{x}^{3}+4x[/latex].
- [latex]h\left(x\right)[/latex] cannot be written in this form and is therefore not a polynomial function.
Licenses & Attributions
CC licensed content, Shared previously
- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..
