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Difference between revisions of "2023 AMC 10A Problems/Problem 21"

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The roots of <math>P(x)</math> are integers, with one exception. The root that is not an integer can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. What is <math>m+n</math>?
 
The roots of <math>P(x)</math> are integers, with one exception. The root that is not an integer can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. What is <math>m+n</math>?
  
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[bold] Solution 1 [\bold]
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Solution 1  
  
 
From the problem statement, we know <math>P(1)=1</math>, <math>P(2-2)=0</math>, <math>P(9)=0</math> and <math>4P(4)=0</math>. Therefore, we know that <math>0</math>, <math>9</math>, and <math>4</math> are roots. Because of this, we can factor <math>P(x)</math> as <math>x(x - 9)(x - 4)(x - a)</math>, where <math>a</math> is the unknown root. Plugging in <math>x = 1</math> gives <math>1(-8)(-3)(1 - a) = 1</math>, so <math>24(1 - a) = 1/24 \implies 1 - a = 24 \implies a = 23/24</math>. Therefore, our answer is <math>23 + 24 =</math> <math>47</math>, or <math>C</math>
 
From the problem statement, we know <math>P(1)=1</math>, <math>P(2-2)=0</math>, <math>P(9)=0</math> and <math>4P(4)=0</math>. Therefore, we know that <math>0</math>, <math>9</math>, and <math>4</math> are roots. Because of this, we can factor <math>P(x)</math> as <math>x(x - 9)(x - 4)(x - a)</math>, where <math>a</math> is the unknown root. Plugging in <math>x = 1</math> gives <math>1(-8)(-3)(1 - a) = 1</math>, so <math>24(1 - a) = 1/24 \implies 1 - a = 24 \implies a = 23/24</math>. Therefore, our answer is <math>23 + 24 =</math> <math>47</math>, or <math>C</math>

Revision as of 19:12, 9 November 2023

Let $P(x)$ be the unique polynomial of minimal degree with the following properties:

  • $P(x)$ has a leading coefficient $1$,
  • $1$ is a root of $P(x)-1$,
  • $2$ is a root of $P(x-2)$,
  • $3$ is a root of $P(3x)$, and
  • $4$ is a root of $4P(x)$.

The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. What is $m+n$?

Solution 1

From the problem statement, we know $P(1)=1$, $P(2-2)=0$, $P(9)=0$ and $4P(4)=0$. Therefore, we know that $0$, $9$, and $4$ are roots. Because of this, we can factor $P(x)$ as $x(x - 9)(x - 4)(x - a)$, where $a$ is the unknown root. Plugging in $x = 1$ gives $1(-8)(-3)(1 - a) = 1$, so $24(1 - a) = 1/24 \implies 1 - a = 24 \implies a = 23/24$. Therefore, our answer is $23 + 24 =$ $47$, or $C$

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~cosinesine


Video Solution 1 by OmegaLearn

https://youtu.be/aOL04sKGyfU

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