Algebra: Elementary to Advanced - Polynomials and Roots from Coursera

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Syllabus

Module 1: Exponentials

The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

Module 2: Polynomials

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Polynomials generalize our linear and quadratic functions that we have studied so far. An example of a polynomial is x^2 − 4x + 7. Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions.

Module 3: Roots

In this module, we will learn about roots of real numbers. Roots arise naturally as solutions to the polynomial equation x^n - a = 0. Square roots help to solve quadratic polynomials. The square root of a nonnegative number is used in the definition of Euclidean distance, as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws. Generalizing square roots lead to n-th roots, their properties, and applications.

Module 4: Applications

Final Exam

Congratulations on reaching the final exam! This final assessment will be cumulative in nature, covering all aspects of the course. Use this final as a teaching tool: justify what you know and identify areas for improvement. Use scrap paper as you take this final. Try to use any formula sheets or outside resources as a tool and not a crutch. Check your answers before you submit. After the test, review any incorrect answers to find your mistakes. Try to separate "silly" mistakes from the more substantial mistakes in understanding. Good luck!

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Covers a variety of topics including exponentials, polynomials, roots, and applications, making it comprehensive for students needing to strengthen their algebra foundation

Helps students develop their logical reasoning and quantitative analysis skills, which are essential for success in higher-level math courses and various fields

Taught by Joseph W. Cutrone, PhD, who has extensive experience and expertise in mathematics education, ensuring the course is well-structured and rigorous

Part of a three-part algebra sequence, providing a solid foundation for students who plan to pursue calculus, statistics, or data science

Utilizes a variety of teaching methods, including videos, readings, and discussions, catering to diverse learning styles and keeping the course engaging

Requires students to have a strong foundation in algebra, making it more suitable for intermediate learners

Reviews summary

Algebra: elementary to advanced - polynomials and roots

Learners say that Algebra: Elementary to Advanced - Polynomials and Roots is an excellent way to refresh algebra skills or begin learning advanced algebra concepts. The course is well structured, easy to navigate, and has a wealth of material including videos, notes, and examples. The course is engaging and well-paced and is taught by an excellent instructor who is knowledgeable and explains concepts clearly. Learners appreciate that assessments have immediate feedback and that each concept is followed by a quiz. Overall, the course is highly recommended and learners say that they have learned a lot.

The course showed how algebraic functions are used in real-world applications.

"I particularly liked the practical application of the algebraic functions."

"An excellent refresher course. This course is subdivided into exponentials, polynomials, roots and applications; all in the context of real numbers."

"I now have the confidence to take additional classes as well as a more analytical mind."

The course had videos, notes, examples, and quizzes.

"Every STEM student should take this course."

"It would be great if you could provide the notes in a PDF format. Thanks."

"All assessments had immediate and comprehensive feedback."

Instructor was knowledgeable and explained concepts well.

"I've really learned a lot in this course and in the whole specialization as a whole. I'm really grateful for the experience and looking forward to taking the introduction to calculus specialization by John Hopkins university."

"I have always been weak in Math and now I know why. Never before did I have an instructor who so clearly explained the concepts behind the formulas and their practical applications."

"Excellent course. Set at a pace and range which helps someone with algebra in life's rear-view mirror get re-engaged in mathematics."

Concepts were explained clearly and systematically.

"This course had a wealth of material that was very easy to navigate."

"I liked the way of explanation very simple and understandable"

"The instructor is fantastic and so knowledgeable!"

The course skipped some material.

"In contrast, the only thing I did not like from this course was it did skip some materiel and information leaving me to go out of my way to learn additional information."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Algebra: Elementary to Advanced - Polynomials and Roots with these activities:

Algebra Review Notebook

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Compiling all your notes, assignments, and practice problems into a comprehensive notebook will serve as a valuable reference and study tool throughout the course.

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Gather all your class notes, homework assignments, and practice problems.
Organize the materials by topic or lesson.
Summarize key concepts and formulas in your own words.
Include additional notes or examples to enhance your understanding.

Linear Equation Refresher

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Reviewing the basics of linear equations will strengthen your foundation and prepare you for more advanced topics in algebra.

Browse courses on Linear Equations

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Review the steps for solving linear equations in one variable.
Practice solving equations with different coefficients and variables.
Identify the slope and y-intercept of linear equations from their equations.

Review Pre-Algebra and Algebra

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By refreshing your knowledge of Pre-Algebra and Algebra, you will be reviewing the fundamentals that this course builds upon.

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Review the order of operations and properties of real numbers.
Practice solving one-step equations.
Review solving multi-step equations.

Eight other activities

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Review: Precalculus by Stewart, 7th Edition

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Reading and reviewing a standard precalculus textbook will provide a comprehensive foundation for the topics covered in this algebra course.

View Notetaking Guide for Stewart's Calculus: Early... on Amazon

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Read the assigned chapters and take notes on key concepts.
Solve practice problems at the end of each chapter to reinforce your understanding.
Summarize the main ideas and formulas in your own words.

Applications of Quadratics Review

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Practicing a variety of quadratic application problems will strengthen your ability to apply algebraic concepts to real-world scenarios.

Browse courses on Quadratic Functions

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Solve word problems involving maximum or minimum values.
Use quadratic equations to model and solve projectile motion problems.
Apply the quadratic formula to find the roots of quadratic equations in context.

Polynomial Graphing Activity

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Creating a program to graph polynomials will reinforce your understanding of their behavior, including things like degree, roots, and asymptotes.

Browse courses on Functions

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Start by choosing a programming language and researching libraries for graphing.
Design a function to take in polynomial coefficients and number of points to evaluate.
Create a loop to evaluate the polynomial at each point and store the result.
Use a library or built-in function to plot the points and connect them with a line.

Exponential Function Practice

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Solving a variety of exponential equations and problems will help you develop a strong foundation in the topic and improve your problem-solving skills.

Browse courses on Exponents

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Find the value of exponential expressions with different bases and exponents.
Solve exponential equations by rewriting them in logarithmic form.
Graph exponential functions and identify their key features, such as intercepts and asymptotes.

Khan Academy: Solving Exponential Equations

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This tutorial provides a step-by-step guide on how to solve exponential equations, which are covered in Module 1 of this course.

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Watch the video tutorial on solving exponential equations.
Complete the practice exercises provided.

Roots of Unity Tutorial

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Following an in-depth tutorial on roots of unity will expand your understanding of complex numbers and their applications in various fields, such as electrical engineering.

Browse courses on Complex Numbers

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Review the basics of complex numbers and polar form.
Learn the definition and properties of roots of unity.
Solve problems involving roots of unity, such as finding their sum or product.

Polynomial Long Division Practice

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Polynomials are covered in Module 2 of this course, and polynomial long division is a key skill for working with polynomials. This drill will give you the practice you need to master this skill.

Browse courses on Polynomials

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Solve 10 polynomial long division problems.
Check your answers using a calculator or online tool.

Algebra Tutoring

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Tutoring other students in algebra will strengthen your own understanding of the subject and develop your communication and teaching skills.

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Offer your services as a tutor to classmates or students in lower-level algebra courses.
Prepare lesson plans and materials to help students understand specific concepts.
Meet with students regularly to provide support and guidance.
Reflect on your tutoring experiences and identify areas for improvement.

Career center

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