Chapter 6 Polynomial Equations

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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

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6.1 Algebra

• Algebra al-jabr (Arabic word meaning
  restoring)
• 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)

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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

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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
5

• 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

x
5
x2
5x
x
25
5x
5
Note we obtained only positive solution!
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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)

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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

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Cardano Formula
substitution x y a/3
sub. y u v
quadratic in u3
roots
9
y u v
Cardano Formula
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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

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substitution x ky
Note
Viète tried to find expressions for cos n? and
sin n? as polynomials in cos ? and sin ?
Newton
Note n is arbitrary (not necessarily integer)
if it is anodd integer the above expression is a
polynomial
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Note Newtons equation has a solution by nth
rootsif n is of the form n4m1 - de Moivre
(1707)
This formula is a consequence of the modern
versionof de Moivres formula
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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
  roots

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
  coefficients

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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
  0
• 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)

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6.8 Biographical Notes Tartaglia, Cardano and
Viète

• 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
  stutterer)
• 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
(Venice)
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• 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.
  Cardano)
• This led to a public contest which was won by
  Ferrari
• 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.

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• 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
  theory
• 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)
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• His family was connected to ruling circles in
  France
• 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
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• 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