# Solving Polynomial Equations

**Babylonians**

Since the first half of the 19^{th} Century, archeologists working in

Mesopotamia have unearthed some half-million inscribed clay

tablets. Many peoples and civilizations, among them the

Sumerians, Akkadians, Chaldeans, and Assyrians have inhabited

the area at one time or another. It is common practice to use the

term Babylonian to refer to all of them, even though the city of

Babylon was only prominent for a short period.

One of several types of quadratic problems that can be found in

the tablets are those of the form:

Given **x + y = a** and **xy = c** find x and y.

Geometrically, such problems are relating the perimeters of

rectangles to their areas, and this gives us a hint as to how they

might have obtained their solutions.

The problems in the clay tablets are, as with the Egyptian papyri,

always ex pressed with specific numbers, and the solutions are

obtained by fol lowing certain algorithms. Examining the algorithms

gives a reasonable explanation of a geometrical approach to their

solutions.

Thus, for the problems of the form x + y = b and xy = c, the

algorithm is:

1. Take the square of ½ b.

2. Subtract c from this value .

3. Take the square root of this value.

4. Add it to ½ b to get x and subtract it from ½ b to get y.

In modern terms, this algorithm is equivalent to what we call the

quadratic formula as taught in high school. We derive the formula

algebraically, but how did the Babylonians come up with it?

No one really knows how the Babylonians arrived at their

algorithm, and several suggestions have been made. Here is mine.

**Problem: **Find a rectangle whose area is c where the sum of the

two sides is b.

**Solution: **By the method of false position.

Let\'s guess that the answer is a square. The square would have

side ½b and area the square of this. **Step 1: Square ½b.**

Since this was a guess, the error made in terms of areas is the

difference between this value and c. **Step 2: Subtract c.**

A geometric square with area equal to this error would have a

side equal to the square root of the area. **Step 3: Take square root.**

We now have to adjust the sides of the square to get a rectangle

that has the right area. The procedure for this correction can be seen

from examining the following diagram.

Thus, the correct dimensions are obtained by adding the
side of

the error square to b/2 and subtracting it from b/2. **Step 4.**

**Greek Solutions**

The Greek geometers would easily find solutions of quadratic

equations geometrically . To solve x^{2} + bx + c = 0, they would **in
principle** find the intersection of the circle y = x

^{2}with the line

y = bx + c by straightedge and compass constructions.

In Euclid\'s Elements, one finds a considerable amount of

discussion on whether or not the solutions are rational. This is

interesting because the issue is of no consequence in the geometric

constructions. It implies that the Greeks were concerned with

numerical solutions, since if the solutions were ir rational they had

no way to represent them other than as lengths of line segments.

**Hindu Solutions
**

There is evidence that the Hindu\'s may have been solving

quadratic equations as early as 200 B.C., but we have no record of

the methods used.

Brahmagupta (c. 628) gives the equation x

^{2}– 10x = -9 and its

solution, showing that algorithms for the solutions of quadratics

were known by this time.

Śrīdhara (c. 1025) was the first, so far as known, to give the socalled

Hindu Rule for quadratics. He is quoted by Bhāskara

(c. 1150) as saying:

**Multiply both sides of the equation by a number equal to four**

times the [coefficient of the] square, and add to them a number

equal to the square of the original [coefficient of the] unknown

quantity. [Then extract the root.]

times the [coefficient of the] square, and add to them a number

equal to the square of the original [coefficient of the] unknown

quantity. [Then extract the root.]

Thus, for the equation ax

^{2}+ bx = c we would get:

These Indian mathematicians worked with both positive and

negative numbers and had a symbol for zero. They worked

effectively with all types of numbers, but usually ignored any

negative roots of equations.

**Al-Khwārizmī**

One of the earliest Islamic algebra texts, written about 825 by

Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) was titled

**Al-kitāb al-muhtaşar fī hisāb al-jabr wa-l-muqābala**

(The condensed book on the calculation of al -Jabr and al-Muqabala)

The term al-jabr can be translated as “restoring” and refers to

transposing a term from one side to the other of an equation. The

word al-muqabala is translated as “comparing” and refers to

subtracting equal quantities from both sides of an equation. Thus

**3x + 2 = 4 – 2x => 5x + 2 = 4** is al-jabr while

**5x + 2 = 4 => 5x = 2 **is al-muqabala.

Our term “algebra” is a corrupted form of the Arabic al-jabr.

Al-Khwārizmī was interested in writing a practical guide to

solving equations, but due to the influence of Greek texts which

were being translated into arabic, he includes geometric

justifications of his manipulations. The geometry however appears

to come from the Babylonians rather than the Greeks.

He classifies the equations he will deal with into six types:

**ax ^{2} = bx ax^{2} = c
bx = c**

ax^{2} + bx = c ax^{2} + c = bx bx + c = ax

^{2 }

The reason for the six-fold classification is that Islamic

mathematicians, un like the Hindus , did not deal with negative

numbers. In their system , coefficients, as well as the roots, had to

be positive. The types listed are the only ones with positive

solutions.

**[ax**

^{2}+ bx + c = 0 would make no sense]**Viète **

François Viète (1540 – 1603) was a French lawyer who worked

for kings Henri III and Henri IV as a cryptanalyst (a breaker of

secret codes). He was so successful at this that his critics

denounced him for being in league with the Devil.

Starting in 1591 he found the time to write several treatises that

are collectively known as The Analytic Art, in which he effectively

reformulated the study of algebra by replacing the search for

solutions with a detailed study of the structure of equations.

Modern symbolism can be traced back to Viete for he was the

first to use letters to represent numbers and manipulate them in

accordance with the rules for manipulating numbers. Without this

idea, his predecessors were forced to consider problems only with

specific numerical coefficients and write out their algorithms in

words.

Viète used vowels to represent unknowns and consonants to

represent given constants. His symbolism was not complete, he still

used words to indicate powers – A^{2} is A quadratum, B^{3} would be B

cubus, and C^{4} is C quadrato-quadratum. He did at times use

abbreviations such as A quad or C quad-quad. His rules for

combining powers had to be given verbally.

The equation x^{3} + cx = d, as we write it today, would have

appeared as

**A cubus + C plano in A aequetus D solido.
**

Note that he used the symbol “+” for addition – he also uses “-” for

subtraction, but the word “in” for multiplication. There is no symbol

for equals (aequetus). The modifiers plano and solido are there to

preserve the law of homogeneity, a Greek concept.

Viete does not yet have the concept of the general quadratic

equation for he gives separate solutions for:

**Si A quad. + B2 in A, aequatur Z plano.**( x

^{2}+ 2bx = c)

**Si A quad. - B in A2, aequatur Z plano.**(x

^{2}– 2bx = c) and

**Si D2 in A – A quad., aequator Z plano.**(2dx – x

^{2}= c)

His solutions are, however, algebraic and produce special cases of

the familiar quadratic formula. Interestingly, in his solutions he

replaces the variable by a sum to obtain the result (today, we call

this a change of variable method) a purely algebraic operation.

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