Introduction to Power and Polynomial Functions: Learn It 3

Polynomial Functions

Now that we’ve explored power functions and seen how a single term can have a powerful impact, let’s take it a step further. Imagine combining multiple power functions into a single, more complex equation. This combination gives rise to polynomial functions, which are incredibly versatile and useful in modeling everything from the trajectory of a rocket to the fluctuations in the stock market.

An oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. The slick is currently [latex]24[/latex] miles in radius, but that radius is increasing by [latex]8[/latex] miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius [latex]r[/latex] of the spill depends on the number of weeks [latex]w[/latex] that have passed. This relationship is linear.

[latex]r\left(w\right)=24+8w[/latex]

We can combine this with the formula for the area [latex]A[/latex] of a circle.

[latex]A\left(r\right)=\pi {r}^{2}[/latex]

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

[latex]\begin{array}{l}A\left(w\right)=A\left(r\left(w\right)\right)\\ A\left(w\right)=A\left(24+8w\right)\\ A\left(w\right)=\pi {\left(24+8w\right)}^{2}\end{array}[/latex]

Multiplying gives the formula below.

[latex]A\left(w\right)=576\pi +384\pi w+64\pi {w}^{2}[/latex]

This formula is an example of a polynomial function.

Polynomial functions allow us to describe curves with multiple peaks and valleys, making them perfect for capturing the intricacies of real-world data.

polynomial functions

Let [latex]n[/latex] be a non-negative integer. A polynomial function is a function that 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]

 

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.

Which of the following are polynomial functions?

[latex]\begin{array}{c}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{array}[/latex]