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)=3x^{7}+2x^{5}-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 2×5 and -3x.

Hence p(x) has at most one positive root.

Now, p(-x)=-3x^{7}-2x^{5}+3x-1=0

Similarly, in p(-x) there are two sign changes.

Hence p(x) has at most two negative roots.

**Example:**

p(x)=2x^{8}+9x^{6}+3x^{2}+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.

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