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>? | ||
− | Solution 1 | + | [bold] Solution 1 [\bold] |
− | From the problem statement, we know P(1)=1, P(2-2)=0, P(9)=0 and 4P(4)=0 | + | |
− | ~aiden22gao | + | 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> |
+ | |||
+ | ~aiden22gao | ||
+ | |||
+ | ~cosinesine | ||
== Video Solution 1 by OmegaLearn == | == Video Solution 1 by OmegaLearn == | ||
https://youtu.be/aOL04sKGyfU | https://youtu.be/aOL04sKGyfU |
Revision as of 19:12, 9 November 2023
Let be the unique polynomial of minimal degree with the following properties:
- has a leading coefficient ,
- is a root of ,
- is a root of ,
- is a root of , and
- is a root of .
The roots of are integers, with one exception. The root that is not an integer can be written as , where and are relatively prime integers. What is ?
[bold] Solution 1 [\bold]
From the problem statement, we know , , and . Therefore, we know that , , and are roots. Because of this, we can factor as , where is the unknown root. Plugging in gives , so . Therefore, our answer is , or
~aiden22gao
~cosinesine