Problem:
Let f be a function for which f(x/3)=x2+x+1. Find the sum of all values of z for which f(3z)=7.
Answer Choices:
A. −1/3
B. −1/9
C. 0
D. 5/9
E. 5/3
Solution:
Let x=9z. Then f(3z)=f(9z/3)=f(3z)=(9z)2+9z+1=7. Simplifying and solving the equation for z yields 81z2+9z−6=0, so 3(3z+1)(9z−2)=0. Thus z=−1/3 or z=2/9. The sum of these values is −1/9.
Note. The answer can also be obtained by using the sum-of-roots formula on 81z2+9z−6=0. The sum of the roots is −9/81=−1/9.
Answer: B.
The problems on this page are the property of the MAA's American Mathematics Competitions