Discuss the maximum possible number of positive and negative roots of

1.	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» 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.

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»

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.