Problem:
Find the number of cubic polynomials , where , and are integers in , such that there is a unique integer with .
Solution:
Answer (738):
Assume that cubic polynomial satisfies the stated conditions. Then for some quadratic polynomial with integer coefficients and coefficient equal to 1 . Because has exactly one integer root different from 2 , either is the square of a linear polynomial for some integer or has 2 as a root and another root different from 2 . Hence all cubic polynomials that satisfy the required conditions are either of the form or of the form for some integers and with .
The coefficient of the linear term has absolute value , so . Because , there are 8 possible choices for , and each of these choices results in a coefficient of the quadratic term whose absolute value does not exceed 20 . For each value of , there are 41 choices for that will make the constant term of fall in the desired range. Therefore in this case, there are such polynomials.
The coefficient of the linear term has absolute value , so . Because , there are 10 possible choices for , and each of these choices results in a coefficient of the quadratic term whose absolute value does not exceed 20. For each value of , there are again 41 choices for that will make the constant term of fall in the desired range. Therefore in this case, there are such polynomials.
Hence there are polynomials satisfying the required conditions.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions