Algebra: Elementary to Advanced - Polynomials and Roots

Go to Course: https://www.coursera.org/learn/polynomials-roots

Introduction

## Course Review: Algebra: Elementary to Advanced - Polynomials and Roots ### Overview The Coursera course titled **"Algebra: Elementary to Advanced - Polynomials and Roots"** is a highly recommended final segment in a comprehensive three-part algebra sequence that pivots from basic principles to intricate mathematical concepts. Whether you’re a student seeking to solidify your algebraic foundation or a professional needing analytical skills for decision-making, this course offers a robust framework to enhance your understanding. One of the standout features of this course is its focus on logical deductions and quantitative analysis—skills that are increasingly vital in today’s data-driven world. The course not only covers advanced functions but also emphasizes modeling and application through algebraic and geometric techniques. This holistic approach cultivates problem-solving skills, making it an excellent resource for anyone aiming to excel academically or professionally. ### Course Syllabus Breakdown The course is structured into four main modules followed by a cumulative final exam, each offering a thoughtful exposition of key algebraic concepts. #### **Module 1: Exponentials** This module begins with a deep dive into exponential functions—heralded as one of the most critical functions in mathematics. You will explore their properties and applications across various fields including physics, biology, and economics. The real-world applications of exponential growth and decay (like compound interest and population dynamics) are elucidated, making it a vital module for understanding complex mathematical phenomena. #### **Module 2: Polynomials** In Module 2, students familiarize themselves with polynomials, which generalize the linear and quadratic functions studied in previous courses. Here, you will learn how to manipulate polynomial expressions and solve polynomial equations, enabling a wide array of applications in science and engineering. The synthesis of theory and practice in this section prepares you to tackle both simple and complex mathematical problems. #### **Module 3: Roots** Module 3 focuses on the concept of roots of real numbers, unraveling their significance in solving polynomial equations. You'll explore square roots as essential components in measuring distances and their implications in statistics, offering insight into topics such as standard deviation. Furthermore, you will learn about n-th roots, which lead to a broader understanding of algebraic structures. #### **Module 4: Applications** The fourth module highlights the practical applications of the topics discussed in the previous sections. By tying together the various concepts of exponentials, polynomials, and roots, you gain an appreciation for how these mathematical fundamentals apply to everyday challenges and advanced scientific problems. #### **Final Exam** The final exam serves as a comprehensive assessment of the material covered. It's designed not only to evaluate knowledge but to enhance learning through reflection on mistakes, making it a valuable learning opportunity. You'll be encouraged to review your solutions and understand the underlying principles, providing an effective way to solidify your grasp of the topics. ### Recommendation I highly recommend the **"Algebra: Elementary to Advanced - Polynomials and Roots"** course for anyone looking to deepen their algebraic understanding. Its structured approach, from exponentials to practical applications, ensures a well-rounded learning experience. With a balance of theoretical insights and hands-on practice, this course empowers you to develop critical analytical skills essential for academic and professional success. In conclusion, whether you are a student aiming to improve your math skills or a professional seeking to enhance your quantitative reasoning, this course is an invaluable resource. Dive in, and you may find your ability to analyze data and make informed decisions significantly sharpened!

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.

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.

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.

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!