Given a polynomial such as:
x^{4}
+ 7x^{3}
 4x^{2}  x  7
it is possible to say anything about how many positive
real roots it has, just by looking at it?
Here's a striking theorem due to Descartes in 1637,
often known as "Descartes' rule of signs":
The number of positive real roots of a polynomial
is bounded by the number of changes of sign in its
coefficients. Gauss later showed that the number
of positive real roots, counted with multiplicity,
is of the same parity as the number of changes of sign.
Thus for the polynomial above, there is at most one positive
root, and therefore exactly one.
In fact, an easy corollary of Descartes' rule
is that the number of negative
real roots of a polynomial f(x) is determined by the
number of changes of sign in the coefficients of f(x).
So in the example above, the number of negative real
roots must be either 1 or 3.
Presentation Suggestions:
Challenge students to prove this fact for
quadratic polynomials.
The Math Behind the Fact:
A proof of Descartes' Rule for polynomials of
arbitrary degree can be carried out by induction.
The base case for degree 1 polynomials is easy to verify!
So assume the p(x) is a polynomial with positive leading
coefficient. The final coefficient of p(x) is given
by p(0).
If p(0)>0, then the number of sign changes must be even,
since the first and last coefficient of p(x) are both
positive. Moreover, the number of roots
(counted with multiplicity) must also be even,
since p(x) is also positive for very large x, so the
graph of p(x) can only cross the xaxis an even number
of times. Similar arguments show that
if p(0)<0, then the number of sign changes is odd and
the number of positive roots is odd. Thus the number
of sign changes and number of roots have the same parity.
If p(x) had more roots than sign changes then it must
have at least 2 more roots. But p'(x) is a polynomial
with zeroes between each of the roots of p(x) [why?],
so p'(x) has at least 1 more root than sign changes of p(x).
This yields a contradiction
because p'(x) has no more sign changes than p(x) does,
and the inductive hypothesis then implies that p'(x)
has no more roots than sign changes of p(x).
How to Cite this Page:
Su, Francis E., et al. "Descartes' Rule of Signs."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
