Write a polynomial function that has 0 possible positive rea

Write a polynomial function that has 0 possible positive real zeros and 5, 3, or 1 possible negative real zero.

Solution

example a polynomial function that has 0 possible positive real zeros and 5,3, or 1 possible negative zero is

f(x)=x⁵+x⁴+x³+x²+x+1

number of sign changes of f(x) =0

f(-x)=-x⁵+x⁴-x³+x²-x+1

number of sign changes of f(-x) =5

by descartes rule of sign function has 0 possible positive real zeros and 5,3, or 1 possible negative zero