2023 AMC 10A Problems/Problem 21: Difference between revisions
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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>? | ||
<math>\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49</math> | |||
==Solution 1== | ==Solution 1== | ||
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== Video Solution 1 by OmegaLearn == | == Video Solution 1 by OmegaLearn == | ||
https://youtu.be/aOL04sKGyfU | https://youtu.be/aOL04sKGyfU | ||
==See Also== | |||
{{AMC10 box|year=2023|ab=A|num-b=20|num-a=22}} | |||
{{MAA Notice}} | |||
Revision as of 20:40, 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 
,
is a root of 
, and
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 
?
Solution 1
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
~walmartbrian
Video Solution 1 by OmegaLearn
See Also
| 2023 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 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 10 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America. 
