Problem:
For some real numbers a and b, the equation
8x3+4ax2+2bx+a=0
has three distinct positive roots. If the sum of the base-2 logarithms of the roots is 5 , what is the value of a ?
Answer Choices:
A. −256
B. −64
C. −8
D. 64
E. 256
Solution:
Let r1​,r2​, and r3​ be the roots. Then
5=log2​r1​+log2​r2​+log2​r3​=log2​r1​r2​r3​
so r1​r2​r3​=25=32. Since
8x3+4ax2+2bx+a=8(x−r1​)(x−r2​)(x−r3​)
it follows that a=−8r1​r2​r3​=−256​.
The problems on this page are the property of the MAA's American Mathematics Competitions