Beyond the Quadratic Formula
MAA, 10 d’oct. 2013 - 228 pàgines
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.
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al-Khwarizmi Cardano’s formula complex conjugate complex numbers Conclude cosine cube roots cubic polynomial x3 Deduce deﬁned deﬁnition Descartes difﬁcult discriminant distinct real roots elementary symmetric polynomials Euler example Exercise factors Ferrari ﬁeld ﬁnd ﬁnding ﬁrst function fundamental theorem geometric given graph imaginary intermediate value theorem irreducible irreducible polynomials Let f(x mathematician mathematics Moivre’s formula multiplication multiplicative inverse nomial non-real complex numbers notation nth roots obtain polynomial equations polynomial expression polynomial f(x polynomial with real positive integer positive real number proof Prove Theorem quadratic equation quadratic formula quadratic polynomial quantity quartic equation quartic polynomial quintic real coefﬁcients reduced cubic equation reduced cubic polynomial resolvent cubic result root of f(x root of q/2 roots are real roots of complex roots of y3 satisﬁes Section solution square roots Suppose symmetric polynomials Tartaglia Theorem 1.4 tion trigonometric turning point Verify write x-axis