Exploration of Functions

1Polynomial Functions

This section is an exploration of polynomial functions, their uses and their mechanics.

1.1Terminology To Know

These are important terms and notations for this section.

1.2What is (and isn’t) a Polynomial?

We know an awful lot about polynomials, but it relies on the very specific structure of a polynomial, and thus it is paramount that one can correctly recognize what is, and isn’t, a polynomial to use these tools.

1.4Fundamental Theorem of Algebra

This section covers one of the most important results in the last couple centuries in algebra; the so-called “Fundamental Theorem of Algebra.”

1.6An Interjection into Polynomial History!

This section is a quick foray into math history, and the history of polynomials!

1.7Exponents and Extrema: An Example

This section contains a demonstration of how odd versus even powers can effect extrema.

1.9Exponents and Extrema 2: Local Extrema

This section contains information on how exponents effect local extrema

1.11Curvature and Graphing

This section shows and explains graphical examples of function curvature.

1.12Factoring: Introduction

Some information on factoring before we delve into the specifics.

1.13Factoring: Common Factors

We discuss step 0 of factoring, extracting any common factors to simplify the process going forward.

1.14Factoring: Round One!

First dive into factoring polynomials. This section covers factoring quadratics with leading coefficient of by factoring the coefficients.

1.16Factoring; Grouping Method

Factor higher polynomials by grouping terms

1.18Factoring; AC Method

How to factor when the leading coefficient isn’t one.

1.20Factoring; Special Forms

Factor polynomials quickly when they are in special forms

1.22Completing the Square

This section introduces the technique of completing the square.

1.24Polynomial Long Division

In this section we explore how to factor a polynomial out of another polynomial using polynomial long division

1.25Polynomial Synthetic Division

Factor one polynomial by another polynomial using polynomial synthetic division

1.27Rational Root Theorem

Find factors via rational root theorem

1.29Complex Numbers

Intro to complex numbers and conjugates

1.30Simplifying Complex Numbers

Simplifying complex numbers

1.32Quadratic Formula

Exploring the usefulness and (mostly) non-usefulness of the quadratic formula

1.33Comprehensive Factoring Quiz

A Comprehensive Factoring Practice Quiz.

2Radical Functions

This section is an exploration of radical functions, their uses and their mechanics.

2.1Terminology To Know

These are important terms and notations for this section.

2.2Why Radicals?

This section introduces radicals and some common uses for them.

2.3Simplifying Radicals - Part 1

This section discusses how to simplify numeric and odd root-value radicals.

2.5Simplifying Radicals - Part 2

This section discusses some of the complications that arise when simplifying radicals.

2.7Square Root: the Inverse Function

This section views the square root function as an inverse function of a monomial. This is used to explain the dreaded symbol and when to use (and not use) absolute values.

2.8Solving Unsimplified Radicals

This section shows techniques to solve an equality that has a radical that can’t be simplified into a non radical form. This has potential drawbacks which is also covered in this section.

3Inequalities

This section is an exploration of inequalities, their uses and their mechanics.

3.1Terminology To Know

These are important terms and notations for this section.

3.2What are Inequalities?

This section introduces inequalities and what they are (and aren’t!)

3.3Computing inequalities

This section discusses how to compute and/or verify inequalities

3.5Algebra with Inequalities

This section discusses how to algebraically manipulate inequalities.

3.6Solving Inequalities

This section discusses how to solve inequalities.

3.8Graphing Inequalities

This section discusses how to graph linear and non-linear inequalities

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