A rational game ends

Algebra Level 4

A polynomial with integer coefficients $P(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots+a_{0}$ , with $a_{m}$ and $a_{0}$ being positive integers, has one of the roots $\frac{2}{3}$ . Find the $n^\text{th}$ smallest $(n \geq 10)$ possible value of $a_{0}+a_{m}$ .

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n+6

n+7

Cannot be determined

n+2

n+5

n+4

n+3