The Chemistry Maths Book, Second Edition

2.5 Polynomials 47

where vis the velocity of the body. The expression for the energy is a quadratic

function of the variables vand x; is the kinetic energy and is the potential

energy. In the absence of external forces the total energy is constant (see Section 12.5

for a more complete discussion of the harmonic oscillator).

EXAMPLE 2.21The simple harmonic oscillator in quantum mechanics

The stationary states of the simple harmonic oscillator in quantum mechanics are

given by the solutions of the time-independent Schrödinger equation

whereψ 1 = 1 ψ(x)is the wave function (see Example 13.11). The equation can be

written in the form

Hψ 1 = 1 Eψ

where the Hamiltonian operator for harmonic motion

is a quadratic function of xand of , since.

The general polynomial

A polynomial of degree ncan always be factorized as the product of nlinear factors

f(x) 1 = 1 a

0

1 + 1 a

1

x 1 + 1 a

2

x

2

1 +1-1+ 1 a

n

x

n

= 1 a

n

(x 1 − 1 x

1

)(x 1 − 1 x

2

)1-1(x 1 − 1 x

n

)

(2.25)

This is called the fundamental theorem of algebra, and was first proved by the great

mathematician Gauss.

5

The function is zero when any of the linear factors is zero, and

the numbersx

1

,x

2

, =,x

n

are the nroots of the polynomial; that is, they are the

solutions of the polynomial equationf(x) 1 = 10. Some of the roots may be equal (multiple

roots) and some may be complex. A polynomial of odd degree (n 1 = 1 1, 1 3, 1 5,1=)

always has at least one real root because its graph must cross the x-axis at least

d

dx

d

dx













=

2

2

2

d

dx

H=− +



22

2

2

2

1

m 2

d

dx

kx

−+=



22

2

2

2

1

m 2

d

dx

kx E

ψ

ψψ

1

2

2

kx

1

2

2

mv

5

Carl Friedrich Gauss (1777–1855), child prodigy and professor at Göttingen, he made substantial contributions

to every important branch of pure and applied mathematics; the theory of numbers, geometry, algebra, statistics,

perturbation theory, electromagnetic theory. He invented or initiated new branches of mathematics, including the

theory of functions of a complex variable and the differential geometry, that formed the basis for much of 19th

century mathematics. He gave his first proof (there were four) of the fundamental theorem of algebra in his doctoral

thesis, Proof of the theorem that every rational integral function in one variable can be resolved into real factors of

first or second degree, 1799.