The Chemistry Maths Book, Second Edition

50 Chapter 2Algebraic functions

Algebraic functions

Polynomials are the simplest examples of algebraic functions. More generally, an

equation of the kind

P(x)y

n

1 + 1 Q(x)y

n− 1

1 +1-1+ 1 U(x)y 1 + 1 V(x) 1 = 10 (2.26)

whereP(x),Q(x), =, V(x)are polynomials of any (finite) degree in x, defines the

variable yas an algebraic function of x. For example, the equation

y

3

1 + 1 (x 1 + 1 1)y

2

1 + 1 (x

2

1 + 13 x 1 + 1 2)y 1 + 1 (x

3

1 + 12 x

2

1 − 1 x 1 − 1 1) 1 = 10

is a cubic equation in y, and can be solved for each value of x.

7

Functions that

cannot be defined in this way in terms of a finite number of polynomials are called

transcendental functions. Examples are the trigonometric functions, the exponential

function, and the logarithmic function; these functions are discussed in Chapter 3.

2.6 Rational functions

LetP(x)andQ(x)be two polynomials

P(x) 1 = 1 a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

1 +1-1+ 1 a

n

x

n

(2.27)

Q(x) 1 = 1 b

0

1 + 1 b

1

x 1 + 1 b

2

x

2

1 +1-1+ 1 b

m

x

m

A rational function, or algebraic fraction, is an algebraic function that has the

general form

(2.28)

Examples of rational functions are

(2.29)

In each case the function is defined for all values of xfor which the denominator is not

zero, since division by zero is not permitted. For example, the function (i) in (2.29) is

not defined atx 1 = 10 , and (iii) is not defined atx 1 = 1 − 2. In general, the rational function

()i ( )ii ( )iii ( )iv

12

1

321

2

1

32

2

2

x

x

x

xx

x

x

x

• 
  

• 
  

+−

• 
  

−

• xx− 1

yfx

Px

Qx

aaxax ax

bbxb

n

n

== =

++ ++

++

()

()

()

01 2

2

01 2



xxbx

m

2 m

++

7

The formula for the general solution of the cubic equation was discovered in Bologna in the early 16th

century by Scipio del Ferro and Nicolo Tartaglia. The method of solution (Cardano’s method) was described by

Girolamo Cardano (1501–1576) in his Ars magnaof 1545. Cardano showed that some solutions are complex. The

book also contains a description of a method of solving quartic equations due to Ludovico Ferrari (1522–1565).

The Norwegian mathematician Niels Henrik Abel (1802–1829) proved in his On the algebraic resolution

of equations(1824) that there does not exist an algebraic solution of the general quintic equation, or of any

polynomial equation of degree greater than 4. ‘Abel’s short life was filled with poverty and tragedy’; he died of

consumption at the age of 27. He gave the first rigorous proof of the binomial theorem, made early contributions

to group theory, and did important and innovative work on the theory of elliptic and other higher transcendental

functions. The general equation of the fifth degree was solved in terms of elliptic functions by Hermite in 1858.