Chapter 6

Polynomial 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 “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 eqationx2 + q = px

• Steps:22

pyx

22

22

pyx

qp

xyyx

22

22 22

2yx

xyyx

Findx and yusing

Note: this is equivalent to the formula

qpp

yx

2

22,

• India, 7th century, Brahmagupta: formula in words expressing general method to solve ax2 + bx = c: a

bbacx

2

4 2

• Greek, Euclid’s “Elements”: rigorous basis for the solution of quadratic equations

• al-Khwārizmī, 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

x2

x

x5x

5x25 5

5

Note: we obtained only positive solution!

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

• 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)

ba

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

Cardano Formula 023 cbxaxx

substitution: x = y – a/3 qpyy 3

sub. y = u + v pyquvyvu

vuuvvuvuy

3)(

)(3)()(33

3333

qvu

puv

33

3

u

pv

3

03

)()(

3

3323

33

puqu

qu

pu

quadratic in u3

roots: symmetry)(by 322

332

3 vpqq

u

323

323

322

322

pqqv

pqqu

qvu 33y = u + v

Cardano Formula:

3

32

3

32

322322

pqqpqqy

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

03 caxx substitution: x = ky cyy 34 3

Note: cos3cos43cos 3 cosy

c3cosViète tried to find expressions for cos nθ and sin nθ as polynomials in cos θ and sin θ

Newton:

sin and sin where!5

)3)(1(

!3

)1( 5222

32

xny

xnnn

xnn

nxy

Note: n is arbitrary (not necessarily integer); if it is anodd integer the above expression is a polynomial

Note: Newton’s equation has a solution by nth rootsif n is of the form n=4m+1 - de Moivre (1707):

nn yyyyx 12

11

2

1 22

This formula is a consequence of the modern versionof de Moivres formula:

nini n sincos)sin(cos

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

0234 dcxbxaxx linear sub. 024 rqxpxx

complete square rpqxpxpx 2222 )(

For any y we have:

)2()2(

)(2)()(222

222222

ypyrpqxxyp

ypxyrpqxpxypx

• 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 money by the letter “K”

• taught himself to read and write

Nicolo Tartaglia (Fontana)1499 (Brescia) – 1557 (Venice)

• 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

Nxxxxxx 45434153 4594595634379545