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 at most 4 positive roots.

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

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

P(x) has almost 3 negative roots.

Since the difference between the number of sign changes in co-efficient P(– x) and the number of negative roots of the polynomial P(x) is even.

The number of negative roots = at most 2.