SOLUTION: Given:{{{ f(x)=3x^5-4x^4+x^3+6x^2+7x-8 }}} How many total zeros does the function have? Possible real zeros? Possible negative zeros?

Question 626899: Given:
How many total zeros does the function have?
Possible real zeros? Possible negative zeros?
Answer by psbhowmick(878) (Show Source): You can put this solution on YOUR website!
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).

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.
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