2004 AMC 12B Problem 17

Attempt Problem 17 of the 2004 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12B solutions, or check the answer key.

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17.

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?

-256

-64

-8

64

256

The sum of

\log_2

of the roots equals

\log_2

of their product

For

8x^3 + \cdots + a,

the product of the roots is

-\dfrac{a}{8}

The sum of the base-

2

logarithms is

\log_2(r_1 r_2 r_3) = 5,

r_1 r_2 r_3 = 2^5 = 32.

By Vieta’s formulas on

8x^3 + 4ax^2 + 2bx + a,

the product of the roots is

-\dfrac{a}{8}.

Thus

-\dfrac{a}{8} = 32,

giving

a = -256.

Thus, the correct answer is A.

Problem 17 in other years