2021 AMC 12B Fall Problem 14

Question

Suppose that

P(z),

Q(z),

and

R(z)

are polynomials with real coefficients, having degrees

2,

3,

and

6,

respectively, and constant terms

1,

2,

and

3,

respectively. Let

N

be the number of distinct complex numbers

z

that satisfy the equation

P(z) \cdot Q(z) = R(z).

What is the minimum possible value of

N?

0

1

2

3

5

Consider

D(z) = P(z)Q(z) - R(z)

; what is its degree and constant term?

A degree-

6

polynomial can have a single root of multiplicity

6

Let

D(z) = P(z)Q(z) - R(z).

Since

P Q

has degree

5

and

R

has degree

6,

the degree of

D

is

6.

Its constant term is

1 \cdot 2 - 3 = -1 \neq 0.

Because

R

is otherwise unconstrained,

D

can be made equal to any real degree-

6

polynomial with constant term

-1,

for instance

-(z - 1)^6.

Such a polynomial has a single distinct root, so the minimum is

N = 1.

Thus, the correct answer is B.

Problem 14 in other years