# The Budan-Fourier Theorem

The Budan-Fourier Theorem

Given a polynomial of degree , the sequence (x)=p(x), (x)=(x), ..., (x)=(x) is called the Budan–Fourier sequence of .

p(x)

n

p

0

p

1

p

p

n

(n)

p

p(x)

Let be the number of real roots of over an open interval (i.e. excluding and ). Then , where is the difference between the number of sign changes of the Budan–Fourier sequence evaluated at and at , and is a non-negative even integer. Thus the Budan–Fourier theorem states that the number of roots in the interval is equal to or is smaller by an even number.

N

p(x)=0

(a,b)

a

b

N=V(a)-V(b)-ν

V(a)-V(b)

a

b

ν

V(a)-V(b)