2020 AMC 12A Problem 15

Question

In the complex plane, let

A

be the set of solutions to

z^3 - 8 = 0

and let

B

be the set of solutions to

z^3 - 8z^2 - 8z + 64 = 0.

What is the greatest distance between a point of

A

and a point of

B?

2\sqrt{3}

6

9

2\sqrt{21}

9 + \sqrt{3}

A

is the three cube roots of

8;

factor the second cubic by grouping

The largest distance pairs a nonreal cube root of

8

with the farthest real root of the second equation

The set

A

consists of the cube roots of

8:

2,

-1 + i\sqrt3,

and

-1 - i\sqrt3.

Factoring by grouping,

z^3 - 8z^2 - 8z + 64

= z^2(z - 8) - 8(z - 8)

= (z - 8)(z^2 - 8),

so

B = \{8, 2\sqrt2, -2\sqrt2\},

all real.

The greatest distance is from

-1 \pm i\sqrt3

to

8:

\sqrt{(8 - (-1))^2 + (\sqrt3)^2}

= \sqrt{81 + 3}

= \sqrt{84} = 2\sqrt{21}.

Thus, D is the correct answer.

Problem 15 in other years