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Chapter 6 Polynomial Equations


Chapter 6 Polynomial Equations Algebra Linear Equations and Eliminations Quadratic Equations Quadratic Irrationals The Solution of the Cubic Angle Division – PowerPoint PPT presentation

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Title: Chapter 6 Polynomial Equations

Chapter 6Polynomial Equations
  • Algebra
  • Linear Equations and Eliminations
  • Quadratic Equations
  • Quadratic Irrationals
  • The Solution of the Cubic
  • Angle Division
  • Higher-Degree Equations
  • Biographical Notes Tartaglia, Cardano and Viète

6.1 Algebra
  • Algebra al-jabr (Arabic word meaning
  • al-Khwarizmi Al-jabr wal mûqabala 830 CE
    (Science of restoring and opposition)
  • restoring adding equal terms to both sides
  • opposing setting the two sides equal
  • Note the word algorithm comes from his name
  • Algebra
  • Indian math inside number theory and
    elementary arithmetic
  • Greek math hidden by geometry
  • Arabic math. recognizes algebra as a separate
    field with its own methods
  • Until the nineteenth century algebra was
    considered as a theory of (polynomial) equations
  • Connection between algebra and geometry analytic
    geometry (Fermat, Descartes, 17th century)

6.2 Linear Equations and Eliminations
  • China (Han dynasty, 206 BCE 220 CE)
    mathematicians invented the method to solve
    systems of linear equations which is now called
    Gaussian elimination
  • They used counting boards to hold the array of
    coefficients and to perform manipulations similar
    to elementary matrix operations
  • Moreover, they discovered that eliminations can
    be applied to polynomial equations of higher
    order in two or more variables

6.3 Quadratic Equations
  • Babylon 2000 BCE algorithm to solve system of
    the form x y p, xy q which is equivalent to
    the quadratic eqationx2 q px
  • Steps

Findx and yusing
Note this is equivalent to the formula
  • India, 7th century, Brahmagupta formula in words
    expressing general method to solve ax2 bx c
  • Greek, Euclids Elements rigorous basis for
    the solution of quadratic equations
  • al-Khwarizmi, 9th century solution, in which
    squares were understood as geometric squares
    and products as geometric rectangles
  • Example solve x2 10x 39
  • x2 and 10x 5x 5x
  • complete the square 25
  • the total area 25 39 64
  • therefore x 5 8 and x 3

Note we obtained only positive solution!
6.4 Quadratic Irrationals
  • Roots of quadratic equations with rational
    coefficients are numbers of the form avb where a
    and b are rational
  • Euclid study of numbers of the form
  • No progress in the theory of irrationals until
    the Renaissance, except for Fibonacci result
    (1225) roots of x32x210x20 are not any of
    Euclids irrationals
  • Fibonacci did not prove that these roots are not
    constructible with ruler and compass (i.e. that
    it is not possible to obtain roots as expressions
    built from rational numbers and square roots)
  • Using field extensions it is not hard to show
    that, say, cube root of 2 is not a quadratic
    irrational and hence is not constructible (and
    this could be done using 16th century algebra)
  • Nevertheless, it was proved only in 19th century
    (Wantzel, 1837)

6.5 The Solution of the Cubic
  • First clear advance in mathematics since the time
    of the Greeks
  • Power of algebra
  • Italy, 16th century Scipione del Ferro, Fior,
    Cardano and Tartaglia
  • Contests in equation solving
  • Most general form of solutionCardano formula

Cardano Formula
substitution x y a/3
sub. y u v
quadratic in u3
y u v
Cardano Formula
6.6 Angle Division
  • France 16th century Viète
  • introduced letters for unknowns
  • and - signs
  • new relation between algebra and geometry
    solution of the cubic by circular (i.e.
    trigonometric) functions
  • his method shows that solving the cubic is
    equivalent to trisecting an arbitrary angle

substitution x ky
Viète tried to find expressions for cos n? and
sin n? as polynomials in cos ? and sin ?
Note n is arbitrary (not necessarily integer)
if it is anodd integer the above expression is a
Note Newtons equation has a solution by nth
rootsif n is of the form n4m1 - de Moivre
This formula is a consequence of the modern
versionof de Moivres formula
6.7 Higher-Degree Equations
  • The general 4th degree (quartic) equation was
    solved by Cardanos friend Ferrari
  • This was solution by radicals, i.e. formula built
    from the coefficients by rational operations and

linear sub.
complete square
For any y we have
  • The r.-h. side Ax2BxC is complete square iff
    B2 - 4AC 0
  • It is a cubic equation in y
  • It can be solved for y using Cardano formulas
  • This leads to quadratic equation for x
  • The final solution for x is a formula using
    square and cube roots of rational functions of

Equations of order 5 and higher
  • For the next 250 years obtaining a solution by
    radicals for higher-degree equations ( 5) was a
    major goal of algebra
  • In particular, there were attempts to solve
    equation of 5th degree (quintic)
  • It was reduced to equation of the form x5 x a
  • Ruffini (1799) first proof of impossibility to
    solve a general quintic by radicals
  • Another proof Abel (1826)
  • Culmination general theory of equations of
    Galois (1831)
  • Hermite (1858) non-algebraic solution of the
    quintic (using transcendental functions)
  • Descartes (1637) (i) introduced superscript
    notations for powers x3, x4, x5 etc. and (ii)
    proved that if a polynomial p(x) has a root a
    then p(x) is divisible by (x-a)

6.8 Biographical Notes Tartaglia, Cardano and
  • spent his childhood in poverty
  • received five serious wounds when Brescia was
    invaded by the French in 1512
  • one of the wounds to the mouth which left him
    with a stutter (nickname Tartaglia
  • at the age of 14 went to a teacher to learn the
    alphabet but ran out of money by the letter K
  • taught himself to read and write

Nicolo Tartaglia (Fontana)1499 (Brescia) 1557
  • moved to Venice by 1534
  • gave public mathematical lessons
  • published scientific works
  • Tartaglia visited Cardano in Milan on March 25,
    1539 and told him about the method for solving
    cubic equations
  • Cardano published the method in 1545 and
    Tartaglia accused him of dishonesty
  • Tartaglia claimed that Cardano promised not to
    publish the method
  • Nevertheless, Cardanos friend Ferrari tried to
    defend Cardano
  • 12 printed pumphlets Cartelli (Ferrari vs.
  • This led to a public contest which was won by
  • Other contribution of Tartaglia to Science
    include a theory describing trajectory of a
    cannonball (which was a wrong theory),
    translation of Euclids Elements (1st
    translation of Euclid in a modern language) and
    translations of some of Archimedes works.

  • Cardano entered the University of Pavia in 1520
  • He completed a doctorate in medicine in 1526
  • became a successful physician in Milan
  • Mathematics was one of his hobbies
  • Besides the solution of the cubic, he also made
    contributions to cryptography and probability
  • In 1570 Cardano was imprisoned by the Inquisition
    for heresy
  • He recanted and was released
  • After that Cardano moved to Rome
  • Wrote The Book of My Life

Giralomo Cardano 1501 (Pavia) 1576 (Rome)
  • His family was connected to ruling circles in
  • Viète was educated by the Franciscans in Fontenay
    and at the University of Poitiers
  • Received Bachelors degree in law in 1560
  • He returned to Fontenay to commence practice
  • Viète was engaged in law and court services and
    related activities and had several very prominent
    clients (including Queen Mary of England and King
    Henry III of France)
  • Mathematics was a hobby

François Viète 1540 - 1603
  • During the war against Spain Viète deciphered
    Spanish dispatches for Henry IV
  • King Philip II of Spain accused the French in
    using black magic
  • Another famous result of Viète was a solution of
    a 45th degree equation posed to him by Adriaen
    van Roomen in 1593
  • Viète recognized the expansion of sin (45 ?) and
    found 23 solutions
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