A rational game 4

Algebra Level 3

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

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The answer is 9.

1 solution

Prince Loomba
Jun 17, 2016

$a_{0}=2x$ and $a_{n}=3y$ because according to rational roots theorem, $2$ is factor of $a_{0}$ and $3$ is factor of $a_{n}$ . Thus we need fourth smallest value of $2x+3y$ , which is obtained when x and y are 3 and 1 respectively. Hence answer is $6+3=9$

Do you have to find all initial maximum values? Is there a better way to solve for maximum values(in order) of this kinda equation?

Akshay Krishna - 2 years, 6 months ago

https://brilliant.org/problems/a-rational-game-ends/ see this problem once!

Prince Loomba - 2 years, 4 months ago

Thanks :) Do you mean induction is the most efficient way?

Akshay Krishna - 2 years, 4 months ago

@Akshay Krishna – I didn't ever say it's the most efficient way... Try to find a better one!! Though I think induction is a good way at least.

Prince Loomba - 2 years, 2 months ago

A rational game 4

For more such problems, click here .

The answer is 9.

1 solution

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