Descartes' Rule of Signs explained by an IITian

This explanation of Descartes' Rule of Signs was submitted by Ayush, a HashLearn tutor from IIT, Dhanbad.

Consider a polynomial equation p(x)=0 (descending powers of x)
i) Number of positive roots of the equation p(x)=0 cannot exceed the number of changes of sign in p(x).
ii)Number of negative roots cannot exceed the number of changes of sign of p(-x).

Example:
p(x)=3x7+2x5-3x-1=0

Clearly, there is only one sign change in p(x) i.e., when sign changes from positive to negative between the terms 2x5 and -3x.
Hence p(x) has at most one positive root.

Now, p(-x)=-3x7-2x5+3x-1=0
Similarly, in p(-x) there are two sign changes.
Hence p(x) has at most two negative roots.

Example: p(x)=2x8+9x6+3x2+7=0
Clearly, here there are no sign changes in p(x) hence it has no positive roots .

Also , there will be no sign changes in p(-x) as all the powers of x are even, hence it will not even have any negative real root.
Thus the given polynomial has no real roots.

You can get similar such explanations from tutors like Ayush on HashLearn. Download and take a session now for free. :)

Download HashLearn

Found it useful? Tag your friends in the comments or share on Facebook! :)