A coprime rational game 3'

Algebra Level 4

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 coprime positive integers , has one of the roots $\dfrac{2}{3}$ . Find the third smallest possible value of $a_{0}+a_{n}$ .

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The answer is 11.

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 $a_{n}$ . One more thing is there that $a_{0} \quad and \quad a_{n}$ are coprime that is x cannot be a multiple of 3 and y cannot be multiple of 2 . Thus we need third smallest value of $2x+3y$ , which is obtained when x and y are 4 and 1 respectively. Hence answer is $8+3=11$

@Prince Loomba

All of the problems in the set are practically based on the same concept, and the same application. I suggest that those problems should be converted into a single problem, and probably another one on the generalization.

Mehul Arora - 4 years, 12 months ago

I did that in one pnc problem, but it became too bad that even I dont want to attempt it. Someone suggested me to make a set. So I did so.

Prince Loomba - 4 years, 12 months ago

Now I suggest you not to make a set of 10 problems out of one idea.

This would simply mean a set named Gamma and asking Gamam 1, Gamma 2, gamma 3 ... Gamma n in n problems,.

Mehul Arora - 4 years, 11 months ago

@Mehul Arora – Thats not same idea. 4 on one idea, 1 on another and 2 on another

Prince Loomba - 4 years, 11 months ago

@Prince Loomba – Still, repeated problems are like CBSE textbooks.

One idea should not be represented in the same way in more than one problem

Mehul Arora - 4 years, 11 months ago

@Mehul Arora – Have you seen such problems before? I am presenting a new use of rational root theorem. To understand my view completely, if anybody wants, therefore I have created similar problems to practice.

Prince Loomba - 4 years, 11 months ago

@Prince Loomba – Yes, there are quite a few problems of the same type.

Mehul Arora - 4 years, 11 months ago

A coprime rational game 3'

For more such problems, click here .

The answer is 11.

1 solution

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