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Polynomial Resolution Theory (Paperback)
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Polynomial Resolution Theory (Paperback)
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This book is the definitive work on polynomial solution theory.
Starting with the simplest linear equations with complex
coefficients, this book proceeds in a step by step logical manner
to outline the method for solving equations of arbitrarily high
degree.
Polynomial Resolution Theory is an invaluable book because of its
unique perspective on the age old problem of solving polynomial
equations of arbitrarily high degree.
First of all Hardy insists upon pursuing the subject by using
general complex coefficients rather than restricting himself to
real coefficients. Complex numbers are used in ordered pair (x, y)
form rather than the more traditional x + iy (or x + jy) notation.
As Hardy comments, "The Fundamental Theorem of Algebra makes the
treatments of polynomials with complex coefficients mandatory. We
must not allow applications to direct the way mathematics is
presented, but must permit the mathematical results themselves
determine how to present the subject. Although practical,
real-world applications are important, they must not be allowed to
dictate the way in which a subject is treated. Thus, although there
are at present no practical applications which employ polynomials
with complex coefficients, we must present this subject with
complex rather than restrictive real coefficients."
This book then proceeds to recast familiar results in a more
consistent notation for later progress. Two methods of solution to
the general cubic equation with complex coefficients are presented.
Then Ferrari's solution to the general complex bicubic (fourth
degree) polynomial equation is presented. After this Hardy
seamlessly presents the first extension of Ferrari's work to
resolving the general bicubic (sixth degree) equation with complex
coefficients into two component cubic equations. Eight special
cases of this equation which are solvable in closed form are
developed with detailed examples. Next the resolution of the octal
(eighth degree) polynomial equation is developed along with twelve
special cases which are solvable in closed form.
This book is appropriate for students at the advanced college
algebra level who have an understanding of the basic arithmetic of
the complex numbers and know how to use a calculator which handles
complex numbers directly.
Hardy continues to develop the theory of polynomial resolution to
equations of degree forty-eight. An extensive set of appendices is
useful for verifying derived results and for rigging various
special case equations.
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