Question

Show that the equation x⁹ – 5x⁵ + 4x⁴ + 2x² + 1 = 0 has atleast 6 imaginary solutions

Answer 1

P(x) = x⁹ – 5x⁵ + 4x⁴ + 2x² + 1

(i) The number of sign changes in P(x) is 2.

The number of positive roots is atmost 2.

(ii) P(– x) = – x⁹ + 5x⁵ + 4x⁴ + 2x² + 1.

The number of sign changes in P(– x) is 1.

The number of negative roots of P(x) is at most 1.

Since the difference of number of sign changes in P(– x) and number of negative zeros is even.

P(x) has one negative root.

(iii) 0 is not the zero of the polynomial P(x).

So the number of real roots is almost 3.

∴ The number of imaginary roots at least 6.