Final answer:
The complex conjugate root theorem suggests that the other two zeros of the polynomial are the conjugates of the given zeros. The end behavior of the polynomial is determined by its highest degree term. The type of extrema depends on the zeros of the first derivative and the change in sign around these points.
Step-by-step explanation:
Complex conjugate root theorem and polynomial division are utilized to answer part a) of the question for finding the remaining zeros of the polynomial p(x). Since we are given that –½ and –1 + 2i are zeros, we know that their conjugates, –½ and –1 - 2i respectively, must also be zeros of p(x) due to the theorem stating that the non-real roots of real polynomials come in conjugate pairs.
For part b), we describe the end behavior of the function using limit notation. The limits as x approaches positive and negative infinity are provided to showcase how the polynomial behaves at extreme values of x. Since the leading term is 2x4, which is positive, as x approaches infinity, so does p(x). Conversely, as x approaches negative infinity, the value of p(x) likewise approaches infinity, since an even power preserves the sign.
Considering the extrema of the polynomial in part c), we understand that extrema occur where the first derivative is zero and changes sign. Since this is a fourth-degree polynomial, we can expect up to three extrema, which could be any combination of local maxima, local minima, or saddle points (points of inflection where the derivative is zero but does not change sign).