5.1: Why It Matters- Polynomials

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Why learn about polynomials?

If you have ever watched a Pixar movie, you have seen computer generated images. A very common method for generating graphics with a computer is to use what is called a wire mesh. You can think of a wire mesh as a grid – like the ones we have used to graph lines – that has been bent and stretched to make a shape we want, as in the image of a dolphin below.

$Dolphin rendered with a mesh of triangles$

The dolphin in the image was created by plotting points in space to create connected triangles. This method of rendering graphics works well and is in wide use, but it takes a lot of computer memory. Recently, researchers have been investigating the use of polynomials for rendering graphics in part because it demands less memory in the process.^[1]. In this process, the surfaces that are rendered are made from solutions to algebraic polynomials. The image below shows some images of smooth-surfaced objects that were rendered using polynomials by researchers Charles Loop and Jim Blinn from Microsoft.

$8 figures rendered with a computer using a polynomial algorithm instead of a triangular mesh. Shapes include a cone, cylinder, sphere, bowling pins, @, a torus and two original shapes.$

Surfaces Rendered Using Polynomials.

In this module, you will learn how to identify a polynomial and how to perform algebraic operations on them. Like the linear equations and inequalities you learned about earlier, polynomials are useful in many applications of mathematics as well as in other disciplines like biology, economics, and even cryptology. Gaining a basic understanding of their qualities and how the rules of algebra we have learned so far apply to them will help you learn how to use polynomials both inside and out of your math class.

In your next math class you will likely learn how to solve certain kinds of polynomials and how to graph them as well.

Learning Outcomes

Single Variable Polynomials
- Define and evaluate polynomials
Operations on Polynomials
- Add polynomials
- Multiply polynomials
- Subtract polynomials
- Multiply binomials
- Divide by a polynomial
Applications of Polynomials
- Divide by a monomial
- Write polynomials involving perimeter, area, and volume
- Write polynomials involving cost, revenue, and profit

Loop, Charles, and Jim Blinn. Real-time GPU Rendering of Piecewise Algebraic Surfaces. ACM SIGGRAPH 2006 Papers on - SIGGRAPH '06 (2006): n. pag. Web. ↵

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