of positive roots a

POLYNOMIAL

equation may possess.

This bound is given by the number of sign changes that

occur when the terms of the polynomial are written in

descending order of degree. As an example, in reading

from left to right, the polynomial equation:

x6+ 7x5– 2x4– 6x3– 7x2+ 8x– 2 = 0

has three sign changes (from positive to negative, nega-

tive to positive, and back again), and so the equation

has at most three positive solutions.

The rule can be extended to count negative roots as

well by replacing “x” with “–x” (which, in effect,

reflects the negative x-axis to the positive side) and

applying the same rule to the polynomial that results.

In the example above, the resultant polynomial is:

(–x)6+ 7(–x)5– 2(–x)4– 6(–x)3– 7(–x)2+ 8(–x) – 2 = 0

that is,

x6– 7x5– 2x4+ 6x3– 7x2– 8x– 2 = 0

That there are four sign changes indicates that there are

at most four negative solutions to the original equation.

As another example, one can quickly check that

the equation x6– 64 = 0 has at most one positive solu-

tion (which must be x= 2), and no negative solutions.

Proof of the Rule

Very few mathematics texts present a proof of

Descartes’s famous result. The argument, unfortunately,

is not elementary and relies on techniques of

CALCULUS

.

We present here a proof that also makes use of the

principle of mathematical

INDUCTION

.

Descartes’s rule of signs certainly works for polyno-

mial equations of degree one: an equation of the form

ax + b= 0, with aand beach different from zero, has

one positive solution if aand bare of different signs,

and no positive solutions if they are the same sign.

Assume Descartes’s rule of signs is valid for any

polynomial equation of degree n, and consider a poly-

nomial p(x) = an+1xn+1 + anxn+ … + a1x+ a0of degree

n+ 1. Its derivative p′(x) = (n+ 1)an+1xn+ nanxn–1 + …

+a1is a polynomial of degree nand so, by assumption,

has at most kpositive roots, where kis the number of

sign changes that occur. Each root of p′(x) represents a

local maximum (hill) or local minimum (valley) of the

original polynomial, and a root of the original polyno-

mial can only occur directly after one such location.

Thus the original polynomial has at most kpositive

roots after the location of the first positive root of

p′(x).Whether the graph of p(x) crosses the positive x-

axis just before this first local maximum or minimum

depends on the signs of p(0) = a0and p′(0) = a1. If

both are positive, then the graph is increasing to a

local maximum just to the right of x= 0, and there is

no additional root. Similarly, there is no additional

root if both are negative. Only if a0and a1have oppo-

site signs could the original equation have k+ 1 rather

than just kpositive roots.

As the sign changes of the derivative p′(x) match

those of p(x),and with the additional consideration of

a possible sign change between a0, and a1, we have that

the number of sign changes of p(x) does indeed match

the number of possible positive roots it could possess.

This proves the rule of signs.

We can further note that if a local maximum to a

graph occurs below the x-axis, or if a local minimum

occurs above the axis, then the polynomial fails to

cross the x-axis twice. Thus, the number of positive

roots a polynomial possesses could miss the number

indicated by the count of sign changes by a multiple of

2. This leads to a more refined version of Descartes’s

rule of signs:

Write the terms of a polynomial from highest

to lowest powers, and let kbe the number of

sign changes that occur in reading the coeffi-

cients from left to right. Then that polynomial

has at most kpositive roots. Moreover, the

number of positive roots it does possess will be

even if kis even, and odd if kis odd.

A bound on the number of negative roots can

be found substituting –xfor xand applying the

same rule to the modified polynomial.

determinant In the study of

SIMULTANEOUS LINEAR

EQUATIONS

, it is convenient to assign to each square

MATRIX

(one representing the coefficients of the terms

of the simultaneous equations) a number called the

determinant of that square matrix. To explain, consider

the simple example of a pair of linear equations:

ax + by = e

cx + dy = f

126 determinant