Question 626899
Every polynomial in one variable of degree n, n > 0, has exactly n real or complex zeros.

Total zeroes = total no. of roots = degree of the polynomial = highest index = 5


Since the coefficients of the polynomial are real so is there has to be complex roots, that will occur in pairs - at max there can be two pairs of complex and conjugate roots. Hence, there is at least one real root.


To find max no. of negative roots, express f(x) as f(-x).

{{{ f(-x)=-3x^5+4x^4-x^3+6x^2-7x-8 }}}

The no. of sign changes from the term with highest degree of x to that with lowest degree of x is 4. Thus max. possible no. of negative roots is 4.