2021 Fall AMC 12B Problem 14

Problem 14 of 25IntermediateAlgebra

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?

Show solution

Solution

Let

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

Since

P Q

has degree

5

and

R

has degree

6,

the degree of

D

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.