A rational game ends

Algebra Level 3

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+2

Cannot be determined

n+7

n+5

n+4

n+3

1 solution

Prince Loomba
Jun 17, 2016

$a_{0}=2x$ and $a_{m}=3y$ because according to rational roots theorem, $2$ is factor of $a_{0}$ and $3$ is factor of $a_{m}$ . Thus we need $n^{th}$ smallest value of $2x+3y$ , and we see that when n=2, answer is 7, n=3 answer 8. Thus all numbers are covered then and it becomes an AP. So for the given condition we can easily write AP as n+5. This I have proved by induction that all numbers $\geq 7$ are covered .

nice solution...+1

Ayush G Rai - 4 years, 12 months ago

A rational game ends

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1 solution

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