DESCARTES RULES OF SIGNS PRACTICE WORKSHEET

Problem 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

Problem 2 :

Discuss the maximum possible number of positive and negative zeros of the polynomials

x² - 5x + 6 and x² - 5x + 16.

Also draw rough sketch of the graphs.

Problem 3 :

Show that the equation

x⁹ - 5x⁵ + 4x⁴ + 2x² + 1 = 0

has atleast 6 imaginary solutions.

Problem 4 :

Determine the number of positive and negative roots of the equation

x⁹ - 5x⁸ - 14x⁷ = 0

Problem 5 :

Find the exact number of real zeros and imaginary of the polynomial

x⁹ + 9x⁷+ 7x⁵ + 5x³ + 3x

Answer Key

1) So, at least 4 positive real roots, at least 3 negative real roots and 2 imaginary roots will be there.

2) Therefore it has atleast 2 positive roots and no negative roots.

3) it has atleast 6 imaginary roots.

4) the given polynomial will have atleast 1 positive root, 1 negative root and 7 imaginary roots.

5) the given polynomial will have no positive, no negative and 9 imaginary roots.