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

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Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties:
  
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*<math>P(x)</math> has a leading coefficient <math>1</math>,
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*<math>1</math> is a root of <math>P(x)-1</math>,
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*<math>2</math> is a root of <math>P(x-2)</math>,
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*<math>3</math> is a root of <math>P(3x)</math>, and
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*<math>4</math> is a root of <math>4P(x)</math>.
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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>?

Revision as of 16:23, 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 $m+n$?

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