2021 Fall AMC 12B Problems/Problem 14

Problem

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 $N$ ?

$\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$

Solution

The answer cannot be $0$ , as every nonconstant polynomial has at least $1$ distinct complex root (fundamental theorem of algebra); the polynomial $P(z)\cdot Q(z)$ has degree $2 + 3 = 5$ , so the polynomial $R(z) - P(z)\cdot Q(z)$ has degree $6$ and is thus nonconstant. It now suffices to illustrate an example for which $N = 1$ . Take $P(z)=z^2+1,Q(z)=z^3+2,$ and $R(z)=(z+1)^6 + \left(z^2+1\right)\left(z^3+2\right).$

$R(z)$ has degree 6 and constant term $3$ , so it satisfies the conditions. We need to find the solutions to $\left(z^2+1\right)\left(z^3+2\right)=(z+1)^6 + \left(z^2+1\right)\left(z^3+2\right),$ or $(z+1)^6=0.$ Clearly, there is one distinct complex root, $-1$ , so our answer is $\boxed{\textbf{(B)} \: 1}.$

~kingofpineapplz and kgator

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2021 Fall AMC 12B Problems/Problem 14

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