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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

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)

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

x

5

x2

5x

x

25

5x

5

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

roots

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

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

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

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

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, 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.

- 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 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