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Chapter 6
Polynomial Equations
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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 “restoring”)
• al-Khwārizmī “Al-jabr w’al 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 eqation
x2 + q = px
2
2
x

y
p
• Steps:
 x y  p
2

2
 x y
 p

  xy     q
 2 
2
2
2

  
 2  2
x y
 x y

xy



2
 2 
2
Note: this is equivalent to the formula
2
p
 p
x, y      q
2
2
Find
x and y
using
• India, 7th century, Brahmagupta:
formula in words expressing
4ac  b 2  b
x
general method to solve
2a
ax2 + bx = c:
• Greek, Euclid’s “Elements”: rigorous basis for the solution
of quadratic equations
–
–
–
–
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!
5x
x2
x
25
5x
• al-Khwārizmī, 9th century: solution, in which “squares”
were understood as geometric squares and “products” as
geometric rectangles
• Example: solve x2 + 10x = 39
5
x
5
6.4 Quadratic Irrationals
• Roots of quadratic equations with rational coefficients are numbers
of the form a+√b where a and b are rational
• Euclid: study of numbers of the form
a b
• No progress in the theory of irrationals until the Renaissance,
except for Fibonacci result (1225): roots of x3+2x2+10x=20 are not
any of Euclid’s 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 solution:
Cardano formula
x 3  ax 2  bx  c  0
Cardano Formula
y 3  py  q
substitution: x = y – a/3
sub. y = u + v
3uv  p
u 3  v3  q
y 3  (u  v)3  (u 3  v 3 )  3uv(u  v) 
 (u 3  v 3 )  3uvy  q  py
3
 p
u    q
 3u 

p
v
3u
3
3
quadratic in u3
2
3
 p
3 2
3
(u )  q(u )     0
3
roots: u 3  q   q    p   v 3 (by symmetry)
2
 2  3 
u 3  v3  q
2
3
2
3
q
q  p
3
u      
2
 2  3 
q
q  p
v      
2
 2  3 
y=u+v
3
Cardano Formula:
2
3
2
q
q
p
q
q
p








3
3
y
     
    
2
2
 2  3 
 2  3 
3
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
x  ax  c  0
3
substitution: x = ky
Note: cos 3  4 cos 3   3 cos 
4 y3  3 y  c
y  cos 
cos 3  c
Viète tried to find expressions for cos nθ and sin nθ
as polynomials in cos θ and sin θ
Newton:
n(n 2  1) 3 n(n 2  1)( n 2  32 ) 5
y  nx 
x 
x 
3!
5!
where y  sin n and x  sin 
Note: n is arbitrary (not necessarily integer); if it is an
odd integer the above expression is a polynomial
Note: Newton’s equation has a solution by nth roots
if n is of the form n=4m+1 - de Moivre (1707):
1n
1n
2
x
y  y 1 
y  y 2 1
2
2
This formula is a consequence of the modern version
of de Moivres formula:
(cos   i sin  ) n  cos n  i sin n
6.7 Higher-Degree Equations
• The general 4th degree (quartic) equation was solved by
Cardano’s friend Ferrari
• This was solution by radicals, i.e. formula built from the
coefficients by rational operations and roots
x 4  ax 3  bx 2  cx  d  0
complete square
linear sub.
x 4  px 2  qx  r  0
( x 2  p) 2  px 2  qx  p 2  r
2
2
2
2
2
2
(
x

p

y
)

(
px

qx

p

r
)

2
y
(
x

p
)

y
For any y we have:
 ( p  2 y) x 2  qx  ( p 2  r  2 py  y 2 )
• The r.-h. side Ax2+Bx+C 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
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)
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
Nicolo Tartaglia (Fontana)
money by the letter “K”
1499 (Brescia) – 1557 (Venice)
• taught himself to read and write
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•
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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, Cardano’s 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 Euclid’s “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 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)
• His family was connected to ruling
circles in France
• Viète was educated by the Franciscans
in Fontenay and at the University of
Poitiers
• Received Bachelor’s 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 θ) and found 23
solutions
45 x  3795 x  95634 x    945 x  45 x  x  N
3
5
41
43
45