Descartes's Rule of Signs

degree of polynomial

1

2

3

4

5

6

7

new polynomial

zoom

1

2

3

4

5

p(x)

3

x

+

2

x

+x+1

3 x	2 x	x	1
1	1	1	1
-1	1	-1	1

sign changes

0

3

Descartes's rule of signs estimates the greatest number of positive and negative real roots of a polynomial

p(x)=

a

n

x

+

a

n-1

x

+⋯+

a

1

x+

a

0

.

Delete any zeros in the list of coefficients and count the sign changes in the new list. If the number of changes is

m

, then the maximum number of positive roots is one of

m

,

m-2

, ….

To get the maximum number of negative roots, use the same procedure for

p(-x)

. The coefficients for

p(x)

and

p(-x)

are shown in the second and third rows of the table.