Answer:
(a) The number of positive zeros is either 2 or 0. The number of negative zeros is either 2 or 0.
Explanation:
The possible numbers of real zeros can be found using Descarte's Rule of Signs.
Rule
When signs of coefficients of terms of the polynomial are listed in order of decreasing degree, the number of sign changes can be counted. That number is the number of possible positive real zeros of the polynomial.
When the signs of odd-degree terms are changed, the number of sign changes becomes the possible number of negative real zeros.
The actual number of real zeros may be less by a multiple of 2. (The zeros may be complex, and complex zeros come in pairs.)
Application
The signs of the terms in the given polynomial are ...
+ - + + + . . . . . 2 sign changes: 2 or 0 positive real zeros
When the signs of odd-degree terms are changed, they become ...
+ + + - + . . . . . 2 sign changes: 2 or 0 negative real zeros
Descarte's rule of signs lets us conclude ...
- The number of positive zeros is either 2 or 0.
- The number of negative zeros is either 2 or 0.
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Additional comment
When we graph the function, we find there are no real zeros. All four of the zeros are complex.