Question

Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x⁹ – 4x⁸ + 4x⁷ – 3x⁶ + 2x⁵ + x³ + 7x² + 7x + 2 = 0.

Answer 1

P(x) = 9x⁹ – 4x⁸ + 4x⁷ – 3x⁶ + 2x⁵ + x³ + 7x² + 7x + 2

The number of sign changes in P(x) is 4. 

∴ P(x) has atmost 4 positive roots. 

P(-x) = -9x⁹ – 4x⁸ - 4x⁷ – 3x⁶ - 2x⁵ - x³ + 7x² - 7x + 2

The number of sign changes in P(-x) is 3. 

Therefore, P(x) has atmost 3 negative roots.