### Instruction: A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?

(A) $\frac{1 + i \sqrt{11}}{2}$

(B) $\frac{1 + i}{2}$

(C) $\frac{1}{2} + i$

(D) $1 + \frac{i}{2}$

(E) $\frac{1 + i \sqrt{13}}{2}$.### Response: I want to find a complex zero of the polynomial that is compatible with the given information.
I know that if a polynomial with real coefficients has a complex zero, then its conjugate must also be a zero.
So, if one of the options is a zero, then its conjugate must also be a zero, and together they must form a quadratic factor of the polynomial.
I also know that the product of the zeros of the polynomial must equal the constant term, by Vieta's formulas.
So, I want to check which option, when multiplied by its conjugate, gives a product that is a factor of the constant term.
I also want to avoid options that would give a quadratic factor with a non-integer coefficient, since that would contradict the given information.
I start with option (A).
Its conjugate is $\frac{1 - i \sqrt{11}}{2}$, and their product is $\frac{1 + i \sqrt{11}}{2} \cdot \frac{1 - i \sqrt{11}}{2} = \frac{1 - 11i^2}{4} = \frac{12}{4} = 3$.
This could be a factor of the constant term, but I don't know what the constant term is.
So, I keep this option as a possibility.
I move on to option (B).
Its conjugate is $\frac{1 - i}{2}$, and their product is $\frac{1 + i}{2} \cdot \frac{1 - i}{2} = \frac{1 - i^2}{4} = \frac{2}{4} = \frac{1}{2}$.
This could not be a factor of the constant term, since the constant term must be an integer.
So, I eliminate this option.
I go to option (C).
Its conjugate is $\frac{1}{2} - i$, and their product is $\frac{1}{2} + i \cdot \frac{1}{2} - i = \frac{1}{4} - i^2 = \frac{1}{4} + 1 = \frac{5}{4}$.
This could also not be a factor of the constant term, for the same reason as before.
So, I eliminate this option.
I check option (D).
Its conjugate is $1 - \frac{i}{2}$, and their product is $1 + \frac{i}{2} \cdot 1 - \frac{i}{2} = 1 - \frac{i^2}{4} = 1 + \frac{1}{4} = \frac{5}{4}$.
This is the same as option (C), so I eliminate it as well.
I look at option (E).
Its conjugate is $\frac{1 - i \sqrt{13}}{2}$, and their product is $\frac{1 + i \sqrt{13}}{2} \cdot \frac{1 - i \sqrt{13}}{2} = \frac{1 - 13i^2}{4} = \frac{14}{4} = \frac{7}{2}$.
This could also not be a factor of the constant term, for the same reason as before.
So, I eliminate this option.
Now, I only have one option left: option (A).
This must be the correct answer, since the problem says that one of the options is a zero of the polynomial.