Modeling Polynomial Functions and Exploring Linear Algebra and Matrices Module Overview

This module is divided into two parts, each dedicated to distinct subjects. Part one delves into polynomial functions, while part two explores the realms of linear algebra an matrices. Both parts will be evaluated independently to gauge your comprehension and development.

Part 1: Polynomial Functions

In mathematics, a polynomial is an expression consisting of variables and coefficients. Polynomial expressions have the following operations: addition, subtraction, multiplication, division, and non-negative integer exponents.

Polynomials have many applications; here are a few of the applications:

1. Finding displacement of objects in Newtonian mechanics, such as how fast or slow it takes an object to fall from a given height, whether it is thrown or dropped. The optimal arch of a basketball as someone shoots it toward the rim. How long, what velocity, and what arch a quarterback needs to throw a football to a receiver.

2. Economists use polynomials to represent cost functions, and they also use them to interpret and forecast market trends. Statisticians use mathematical models, where polynomials are used to analyze and interpret data, as well as draw conclusions from the data. Financial planners use polynomials to calculate interest rate problems to determine how much money a person needs to accumulate over a given amount of time with a specified initial investment.

3. In meteorology, polynomials are used to create mathematical models that represent weather patterns that can be analyzed to make weather predictions.

4. Engineers that design roller coasters use polynomials to describe the various curves in the rides.

Picture yourself riding the space shuttle to the international space station. You will need to calculate your speed so you can make the proper adjustments to dock with the station. Or you are on the design team for the US Olympic speed cycling event and you need to correct a flaw in the wheel balance of the cycles. Or you are starting your own business and want to ensure that you have maximum profit. Each of these situations involves the use of polynomials, which are mathematical expressions with many (poly-) terms. Polynomials are used to represent an amazing number of real world situations like the ones above, as well as in photography, sales, advertising, design, pollution and data analysis, to name just a few.

Essential Questions

How is the imaginary number i defined?
How are complex numbers defined?
What are the rules for complex numbers operations?
What is a quadratic function?
What are the rules for graphing quadratic functions?
What are the characteristics of a quadratic graph and how are they represented?
How are "zeros" of polynomial functions computed?
What is the Remainder Theorem?
What are the types of Polynomial Functions?
What are some key features of Polynomial Graphs?

Polynomial Functions Key Terms

Polynomial - The sum or difference of two or more monomials.

Constant - A term with degree 0 (a number alone, with no variable).

Monomial - An algebraic expression that is a constant, a variable, or a product of a constant and one or more variables (also called "terms").

Binomial - The sum or difference of two monomials.

Trinomial - The sum or difference of three monomials.

Degree of the Polynomial - The largest sum of the exponents of one term in the polynomial.

Integers - Positive, negative and zero whole numbers (no fractions or decimals).

Like Terms - Terms having the exact same variable(s) and exponent(s).

Coefficient - Number factor; number in front of the variable.

Imaginary Number - A number that involves i which is equation image indicator $LaTeX: \sqrt[]{-1}$ $\sqrt[]{-1}$ .

Complex Number - A number with both a real and an imaginary part, in the form equation image indicator LaTeX: a+bi $a+bi$ .

Conjugate - The same binomial expression with the opposite sign.

Greatest Common Factor - Largest expression that will go into the terms evenly.

Zeros - The roots of a function, also called solutions or x-intercepts.

Constant - A "0" power (degree) polynomial.

Linear - A 1^st power (degree) polynomial.

Quadratic - A 2^nd power (degree) polynomial.

Cubic - A 3^rd power (degree) polynomial.

Quartic - A 4^th power (degree) polynomial.

Quintic - A 5^th power (degree) polynomial.

Intercepts - Points where a graph crosses an axis.

System of Equations - n equations with n variables.

Point of Intersection - The point(s) where the graphs cross.

Consistent - Has at least one solution.

Inconsistent - Has no solution.

Domain - The values for the x variable.

Range - The values for the y variable.

Extrema - Maximums and minimums of a graph.

Part 2: Exploring Linear Algebra and Matrices

We will explore different ways to portray math that you may already know. In this Module, we will discover Matrices. Matrices can be used to display and organize data. When data is organized in a matrix, you can add, subtract and even multiply the data to manipulate it! You will come to see that matrices have many of the same properties as real numbers, except when multiplied! Not only can you display data with matrices, you can also solve systems of equations and transform geometric figures.

Essential Questions

How can we represent data in matrix form?
How do we add, subtract and multiply matrices?
What is an identity matrix and how does it behave?
How do we find the determinant and the inverse of a 2x2 matrix?
How do we solve systems of equations using inverse matrices?
How do we find the area of a parallelogram using matrices?
How do you work with matrices as transformations in the plane?

Matrices Key Terms

The following key terms will help you understand the content in this module.

Determinant - the product of the elements on the main diagonal minus the product of the elements of the other diagonal (for a 2 x 2 matrix)

Dimensions of a Matrix - the number of rows by the number of columns

Identity Matrix - a square matrix with 1's on the main diagonal and 0's everywhere else

Matrix - a rectangular arrangement of numbers into rows and columns

Scalar - in matrix algebra, a real number is called a scalar

Square Matrix - a matrix with the same number of rows and columns

Vector - a quantity that has both magnitude and direction

Zero Matrix - a matrix whose entries are all zeros

IMAGES CREATED BY GAVS