Beyond the Quadratic Formula

Beyond the Quadratic Formula

Ron Irving
Copyright Date: 2013
Edition: 1
Pages: 245
https://www.jstor.org/stable/10.4169/j.ctt5hh9sp
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  • Book Info
    Beyond the Quadratic Formula
    Book Description:

    The quadratic formula for the solution of quadratic equations was discovered independently by scholars in many ancient cultures and is familiar to everyone. Less well known are formulas for solutions of cubic and quartic equations whose discovery was the high point of 16th century mathematics. Their study forms the heart of this book, as part of the broader theme that a polynomial’s coefficients can be used to obtain detailed information on its roots. A closing chapter offers glimpses into the theory of higher-degree polynomials, concluding with a proof of the fundamental theorem of algebra. The book also includes historical sections designed to reveal key discoveries in the study of polynomial equations as milestones in intellectual history across cultures. Beyond the Quadratic Formula is designed for self-study, with many results presented as exercises and some supplemented by outlines for solution. The intended audience includes in-service and prospective secondary mathematics teachers, high school students eager to go beyond the standard curriculum, undergraduates who desire an in-depth look at a topic they may have unwittingly skipped over, and the mathematically curious who wish to do some work to unlock the mysteries of this beautiful subject.

    eISBN: 978-1-61444-112-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Preface
    (pp. ix-xiv)
  3. Table of Contents
    (pp. xv-xvi)
  4. 1 Polynomials
    (pp. 1-20)

    We will be studying polynomial equations throughout this book, especially those of degrees 2, 3, and 4. In this chapter, we introduce terminology and obtain some basic results that hold for all polynomials. The reader familiar with this material may wish to skip ahead. Others may wish on first encounter to read through this chapter’s definitions and results, returning for a closer reading as they are used.

    What is a polynomial? We know that

    $3{x^2} - 4x - 7$

    is one, as is

    $5{x^{17}} + 12{x^{11}} - 4{x^7} + 13{x^4} + {x^3} - x + 113.$

    So are

    $_3^4{x^{100}}|{\rm{ }}_5^2{x^{88}}5$.

    and

    ${\sqrt {2x} ^{333}} - 2{x^{200}} + \frac{1}{3}{x^{111}} + {x^4} - {\pi ^2}.$

    But

    ${x^4} + \sin x$

    is not a polynominal, and neither is

    ${10^x}.$

    (Why not? For now, we...

  5. 2 Quadratic Polynomials
    (pp. 21-46)

    The heart of this book is the study of solutions to cubic and quartic equations, which we will begin in Chapter 3. This chapter is devoted to quadratic equations. Even though they are familiar from a first algebra course, a close look is warranted, as a warmup before we tackle the greater difficulties of cubic and quartic equations and to introduce themes that will recur as we study cubics and quartics.

    The general quadratic equation has the form

    $a{x^2} + bx + c = 0,$

    for real numbersa,b, andc, with$a \ne 0.$. Thequadratic formulafor the solutions of this equation equation takes the...

  6. 3 Cubic Polynomials
    (pp. 47-72)

    In this chapter, we will take our first look at cubic equations and the famous formula for their solution known as Cardano’s formula. Girolamo Cardano, for whom the formula is named, was a sixteenth-century Italian scholar. The story of the formula’s discovery is complex, as we will see in Section 3.5, and credit must be shared with Scipione del Ferro and Niccolò Fontana.

    Our results in this chapter will be imprecise, because we are lacking what turns out to be an essential tool: complex numbers. However, it is this first look that will reveal the need for complex numbers. After...

  7. 4 Complex Numbers
    (pp. 73-108)

    Our experience with Cardano’s formula has taught us that to solve cubic equations, we need to work with numbers such as$\sqrt { - 3} $, that is, square roots of negative numbers. Cubic equations were where they were first encountered. They are now called complex numbers, and have been found to have many important uses in mathematics and science. In this chapter, we will introduce them and see how to calculate theirnth powers and roots, which is needed in our study of polynomial equations.

    In this section we will develop the arithmetic of complex numbers. We begin by introducing a new number...

  8. 5 Cubic Polynomials, II
    (pp. 109-142)

    We derived Cardano’s formula for roots of reduced cubic polynomials in Section 3.2, only to discover that using it may require us to compute cube roots of complex numbers. This phenomenon arose in Exercise 3.12, when we tried to solve the cubic equation${y^3} - 7y + 6 = 0$. Our study of the discriminant in Section 3.4 revealed that this difficultywill occur whenever we work with a cubic whose roots are real and distinct. With that, we brought Chapter 3 to a close and turned to a study of complex numbers in Chapter 4. Now that we have learned how to use trigonometry to compute...

  9. 6 Quartic Polynomials
    (pp. 143-178)

    We learned in Section 3.5 that Lodovicio Ferrari discovered a way to solve quartic equations not long after Cardano’s work on cubic equations, and that Cardano presented Ferrari’s method inArs Magna[12]. We will take a brief look at Ferrari’s approach, then turn to the approach of Renè Descartes. Both reduce the solving of a reduced quartic equation to the determination of the roots of an auxiliary cubic polynomial. A closer look at Descartes’ solution will allow us to obtain a formula due to Euler that expresses the roots of the reduced quartic in terms of the roots of...

  10. 7 Higher-Degree Polynomials
    (pp. 179-216)

    We have devoted chapters to quadratic, cubic, and quartic polynomials. This pattern cannot continue through all degrees, and not just because the resulting book would be infinitely long. It turns out that results of the sort we have obtained do not exist for polynomials in degree greater than four. Therefore, we will content ourselves with a survey of some central results about higher-degree polynomials, combining proof sketches (or no proofs at all) with historical discussions. The chapter ends with a proof of the fundamental theorem of algebra.

    We have obtained the quardic formula, Cardano’ formula, and Euler’s formula for solutions...

  11. References
    (pp. 217-222)
  12. Index
    (pp. 223-226)
  13. About the Author
    (pp. 227-228)