Use descartes rule of signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function: f(x)= 9x^7

Question

Use descartes rule of signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function:
f(x)= 9x^7 + 3x^5 - 8x^3 + 7x - 1

Answer 1

f(x) has 3 variations of sign, so there are either 3 or 1 positive real zeros. (Descartes Rule says that the number of positive real zeros of a polynomial function y = f(x) is equal to the number of sign variations of the coefficients of f(x) or is less than that by an even integer)

 

f(-x) = 9(-x)⁷ + 3(-x)⁵ - 8(-x)³ + 7(-x) - 1

 

        = -9x⁷ - 3x⁵ + 8x³ - 7x - 1

 

f(-x) has 2 variations of sign, so there are either 2 negative real zeros or none at all. (Descartes Rule says that the number of negative real zeros of a polynomial function y = f(x) is equal to the number of sign variations in the coefficients of f(-x) or is less than that by a positive integer)