Show that the polynomial px 2x3 5x 19 has exactly one rea

Show that the polynomial p(x) = 2x^3 + 5x - 19 has exactly one real root.

Solution

By Descartes Rule of signs

p(x) has at most 1 positive real root as there is only one sign change in coefficients

p(-x)=-2x^3-5x-19

So there is no sign change hence no real negative root

Hence it can have at most 1 real root

And complex roots always occur in pair because if z is a complex root then its complex conjugate z\' is also a root

Hence, p(x) can have only 2 complex roots

SO it must have 1 and exactly 1 real root