1. ## polynomial equation (?)

I've just started year 1 at the university, can't really understand what i have to do, the new signs also confuse me, please guide me and show me how i should handle this question:

i think instead of "k" he ment "n"...

Thanks!

2. Originally Posted by kubebm
I've just started year 1 at the university, can't really understand what i have to do, the new signs also confuse me, please guide me and show me how i should handle this question:

i think instead of "k" he ment "n"...

Thanks!

You have to know (and it's easy to prove, but it requires a little abstraction and, perhaps, a little experience in these things) the following:

if $\displaystyle \frac{a}{b}$ is a rational root of the integer polynomial $\displaystyle a_nx^n+a_{n-1}b^{n-1}+...+a_1x+a_0$, then $\displaystyle a\mid a_0\,\,and\,\, b\mid a_n$, with $\displaystyle x\mid y$ meaning "x divides y".

Well, in your case you have $\displaystyle a_0=35^M\,,\,\,a_n=6^N$, so if $\displaystyle \frac{a}{b}$ is a rational root of that pol., it must be that $\displaystyle a\mid 35^M\,\,and\,\,b\mid 6^N\Longrightarrow a\in \pm \{1,5,7,5^2,7^2,...,35, 35^2,...,35^M\}$ $\displaystyle b\in \pm \{1,2,3,2^2,3^2,...,6,6^2,...,6^N\}$

Tonio

3. ## ~

ok thank you very much! but i still dont get it, how i suppose to present the solution?? a means the
coefficient of the last part and b is the one for the first part? do i need to divide each value from group
a in each value of group b and find if it's rational root or what?

4. Originally Posted by kubebm
ok thank you very much! but i still dont get it, how i suppose to present the solution?? a means the
coefficient of the last part and b is the one for the first part? do i need to divide each value from group
a in each value of group b and find if it's rational root or what?

You have to take each value from each group and divide them and that way get the MAXIMAL possible number of rational solutions which is what they're asking you to do.

Tonio