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 $N?$

$0$

$1$

$2$

$3$

$5$

Solution:

Let $D(z) = P(z)Q(z) - R(z).$ Since $P Q$ has degree $5$ and $R$ has degree $6,$ the degree of $D$ 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.

2021 AMC 12B Fall Problem 14

Solution:

Problem 14 in Other Years