Problem:
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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If and are the integer zeros, the polynomial can be written in the form
The coefficient of , is an integer, so is an integer. The coefficient of , is an integer, so is also an integer. Applying the quadratic formula gives the remaining zeros as
Answer choices (A), (B), (C), and (E) require that , which implies that the imaginary parts of the remaining zeros have the form . This is true only for choice .
Note that choice (D) is not possible since this choice requires , which produces an imaginary part of the form , which cannot be .
The problems on this page are the property of the MAA's American Mathematics Competitions