Determine the number of positive and negative roots of the equation x9

1.	Determine the number of positive and negative roots of the equation x9 – 5x8 – 14x7 = 0.
Answer» x⁹ – 5x⁸ – 14x⁷ = 0 P(x) = x⁹ – 5x⁸ – 14x⁷. The number of sign changes is P(x) is 1. The number of positive roots is 1. P (-x) = -x⁹ – 5x⁸ + 14x⁷ The number of sign changes is P(-x) is one. The number of negative zero of P(-x) is 1. It is clear that 0 is a root of the equation. ∴ The number of the imaginary roots is at least 6.

Determine the number of positive and negative roots of the equation x9 – 5x8 – 14x7 = 0.

Answer»

x⁹ – 5x⁸ – 14x⁷ = 0

P(x) = x⁹ – 5x⁸ – 14x⁷.

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

The number of positive roots is 1. P (-x) = -x⁹ – 5x⁸ + 14x⁷

The number of sign changes is P(-x) is one. The number of negative zero of P(-x) is 1. It is clear that 0 is a root of the equation.

∴ The number of the imaginary roots is at least 6.