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### Mathematical tricks for Problem Solving

#### MATH TRICK-10

This is from the chapter of Theory Of Equations.

DESCARTE'S RULE OF SIGNS
(1) The maximum number of positive real roots of a polynomial equation f(x)=0 is the number of changes of signs from positive to negative & negative to positive in f(x).
(2) The number of negative roots of f(x)=0 is the number of changes of signs in f(-x)=0.

USEFUL THEOREM:- Let f(x) be a real polynomial of degree n(≥1) and a,b be two real numbers such that a<b.
(1) If f(a) & f(b) are of positive signs,then polynomial f(x) has at least one and always an odd number of real zeros in (a,b).
(2) If f(a) & f(b) are of the same sign,then polynomial f(x) has either one real zero or an even number of real zeros in (a,b).

Remark:- Every odd degree polynomial has at least one real root since complex roots occur in pairs.

Ex.1.(FROM RMO PAPER)-
If a>0,prove that x^3 +ax+b =0 has two complex roots.

Sol:- Case-1: b>0
Let f(x)=x^3 +ax+b
Signs are  +,+,+ .
Since there is no sign change in f(x).
Thus,f(x)=0 has no positive root.
Again, f(-x)=-x^3 -ax+b
Signs are -,-,+