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=log2r1+log2r2+log2r3=log2r1r2r3
so r1r2r3=25=32. Since
8x3+4ax2+2bx+a=8(x−r1)(x−r2)(x−r3)
it follows that a=−8r1r2r3=−256.
The problems on this page are the property of the MAA's American Mathematics Competitions