jmc

Question

For some real numbers a and b, the equation 8x^3 + 4ax^2 + 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?

Accepted Answer

Let r_1, r_2, and r_3 be the roots. Then 5= _2r_1 + _2 r_2 + _2 r_3 = _2r_1r_2r_3, so r_1r_2r_3 = 2^5 = 32. Since 8x^3+4ax^2+2bx+a=8(x-r_1)(x-r_2)(x-r_3), it follows that a = -8r_1r_2r_3= -256.