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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 14"

(Created page with "==Problem== Suppose that <math>P(z), Q(z)</math>, and <math>R(z)</math> are polynomials with real coefficients, having degrees <math>2</math>, <math>3</math>, and <math>6</mat...")
 
m (→‎Solution)
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==Solution==
 
==Solution==
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The answer can not be <math>0</math>, as every polynomial has at least <math>1</math> distinct complex root (fundamental theorem of algebra).
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The answer cannot be <math>0</math>, as every nonconstant polynomial has at least <math>1</math> distinct complex root (fundamental theorem of algebra); the polynomial <math>P(z)\cdot Q(z)</math> has degree <math>2 + 3 = 5</math>, so the polynomial <math>R(z) - P(z)\cdot Q(z)</math> has degree <math>6</math> and is thus nonconstant. It now suffices to illustrate an example for which <math>N = 1</math>.
 
Take <math>P(z)=z^2+1,Q(z)=z^3+2,</math> and <math>R(z)=(z+1)^6 + \left(z^2+1\right)\left(z^3+2\right).</math>
 
Take <math>P(z)=z^2+1,Q(z)=z^3+2,</math> and <math>R(z)=(z+1)^6 + \left(z^2+1\right)\left(z^3+2\right).</math>
  
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~kingofpineapplz
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~kingofpineapplz and kgator
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2021 Fall|ab=B|num-a=13|num-b=11}}
 
{{AMC12 box|year=2021 Fall|ab=B|num-a=13|num-b=11}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:09, 16 January 2022

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

See Also

2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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