JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Position of Roots

(1) If \[f(x)=0\] is an equation and \[a,b\] are two real numbers such that \[f(a).f(b)<0\] has at least one real root or an odd number of real roots between \[a\] and \[b\]. In case \[f(a)\] and \[f(b)\] are of the same sign, then either no real root or an even number of real roots of \[f(x)=0\]lie between a and b.

(2) Every equation of an odd degree has at least one real root, whose sign is opposite to that of its last term, provided the coefficient of the first term is \[+ve\] e.g., \[{{x}^{3}}-3x+2=0\] has one real negative root.

(3) Every equation of an even degree whose last term is \[-ve\]  and the coefficient of first term \[+ve\] has at least two real roots, one \[+ve\] and one \[-ve\] e.g., \[{{x}^{4}}+4{{x}^{3}}+3{{x}^{2}}+5x-2=0\] has at least two real roots, one \[+ve\] and one \[-ve\].

(4) If an equation has only one change of sign, it has one \[+ve\] root and no more.

(5) If all the terms of an equation are \[+ve\] and the equation involves no odd power of \[x,\] then all its roots are complex.