1. ## Rational numbers problem

Which rational numbers $\displaystyle t$ are such that
$\displaystyle 3t^3+10t^2-3t$
is an integer?

2. $\displaystyle 3t^3+10t^2-3t= t(3t^2+10t-3)$

Now solve the expression inside the brackets using the quadratic formula.

Do you know this formula?

3. Yes, I know this formula. I solved this problem few minutes ago. I will show my solution tomorrow. It is a little different from this above and not so simple.
Maybe I am wrong.

4. Here is my solution:

It isn't so simple like this two posts above. Is it correct?

5. Originally Posted by Arczi1984
Here is my solution:

It isn't so simple like this two posts above. Is it correct?
Why have you discounted the cases $\displaystyle m=\pm 1$

CB

6. In these two cases we should solve two simple equations like pickslides noted. So I omitted them in my solution.

7. Originally Posted by Arczi1984
In these two cases we should solve two simple equations like pickslides noted. So I omitted them in my solution.
It would make things a bit clearer if you say something like, let $\displaystyle t=n/m$ be a rational number such that $\displaystyle 3t^3+10t^2-3t$ is an integer.

Now the cases $\displaystyle m=\pm 1$ correspond to $\displaystyle t$ being an integer, but $\displaystyle 3t^3+10t^2-3t$ is an integer when ever $\displaystyle t$ is an integer so all integer values of $\displaystyle t$ are trivially among those which make the expression in question an integer.

Or am I still misunderstanding something?

CB

8. Yes, You're right.