2019 AMC 12A Problem 14

Question

For a certain complex number

c,

the polynomial

\begin{aligned} P(x) &= (x^2 - 2x + 2) \\ &\quad {}\cdot (x^2 - cx + 4) \\ &\quad {}\cdot (x^2 - 4x + 8) \end{aligned}

has exactly

4

distinct roots. What is

|c|?

2

\sqrt{6}

2\sqrt{2}

3

\sqrt{10}

The first and third factors give roots

1 \pm i

and

2 \pm 2i

For only

4

distinct roots,

x^2 - cx + 4

must reuse two of these; its roots multiply to

4

The factors

x^2 - 2x + 2

and

x^2 - 4x + 8

have roots

1 \pm i

and

2 \pm 2i,

which are

4

distinct values.

For

P

to have exactly

4

distinct roots, the roots of

x^2 - cx + 4

must lie among these. Their product must equal

4,

and the only such pair is one root from each factor, for example

(1 + i)(2 - 2i) = 4.

Then

c = (1 + i) + (2 - 2i) = 3 - i,

so

|c| = \sqrt{3^2 + 1^2} = \sqrt{10}.

Thus, the correct answer is E.

Problem 14 in other years