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:
x
6 + 7x
5 – 2x
4 – 6x
3 – 7x
2 + 8x – 2 = 0
has three sign changes (from positive to negative, negative 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,
x
6 – 7x
5 – 2x
4 + 6x
3 – 7x
2 – 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 x
6 – 64 = 0 has at most one positive solution (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 polynomial equations of degree one: an equation of the form
ax + b = 0, with a and b each different from zero, has
one positive solution if a and b are 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 polynomial p(x) = a n+1 x
n+1 + a n x
n + … + a 1 x + a 0 of degree
n + 1. Its derivative p′(x) = (n + 1)a n+1 x
n + na n x
n–1 + …
+a 1 is a polynomial of degree n and so, by assumption,
has at most k positive roots, where k is 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 polynomial can only occur directly after one such location.
Thus the original polynomial has at most k positive
roots after the location of the first positive root of
p′(x). Whether the graph of p(x) crosses the positive xaxis just before this first local maximum or minimum
depends on the signs of p(0) = a 0 and p′(0) = a 1 . 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 a 0 and a 1 have opposite signs could the original equation have k + 1 rather
than just k positive 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 a 0 , and a 1 , 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 k be the number of
sign changes that occur in reading the coefficients from left to right. Then that polynomial
has at most k positive roots. Moreover, the
number of positive roots it does possess will be
even if k is even, and odd if k is odd.
A bound on the number of negative roots can
be found substituting –x for x and 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
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:
x
6 + 7x
5 – 2x
4 – 6x
3 – 7x
2 + 8x – 2 = 0
has three sign changes (from positive to negative, negative 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,
x
6 – 7x
5 – 2x
4 + 6x
3 – 7x
2 – 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 x
6 – 64 = 0 has at most one positive solution (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 polynomial equations of degree one: an equation of the form
ax + b = 0, with a and b each different from zero, has
one positive solution if a and b are 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 polynomial p(x) = a n+1 x
n+1 + a n x
n + … + a 1 x + a 0 of degree
n + 1. Its derivative p′(x) = (n + 1)a n+1 x
n + na n x
n–1 + …
+a 1 is a polynomial of degree n and so, by assumption,
has at most k positive roots, where k is 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 polynomial can only occur directly after one such location.
Thus the original polynomial has at most k positive
roots after the location of the first positive root of
p′(x). Whether the graph of p(x) crosses the positive xaxis just before this first local maximum or minimum
depends on the signs of p(0) = a 0 and p′(0) = a 1 . 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 a 0 and a 1 have opposite signs could the original equation have k + 1 rather
than just k positive 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 a 0 , and a 1 , 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 k be the number of
sign changes that occur in reading the coefficients from left to right. Then that polynomial
has at most k positive roots. Moreover, the
number of positive roots it does possess will be
even if k is even, and odd if k is odd.
A bound on the number of negative roots can
be found substituting –x for x and 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