Thread: polynomials

I have one more problem...
I want to find all rings Zn, such the polynomial x+2 divides polynomial (x^5)-10x+12.
Could anybody help me?
Thanks, this is a new topic for me...

Originally Posted by sidi

I have one more problem...
I want to find all rings Zn, such the polynomial x+2 divides polynomial (x^5)-10x+12.
Could anybody help me?
Thanks, this is a new topic for me...

$\displaystyle x^5 - 10x + 12$ is divisible by $\displaystyle x+2$ in any (unitary) ring because $\displaystyle (-2)^5 -10 \times (-2) + 12 = \text{what} $?

of course, -2 is root of that polynomial...I know. I think it should work, that it is divisible in any ring, but I can´t find the reason...
Zn isn´t a subring of Z, or am I wrong??

Originally Posted by sidi

Zn isn´t a subring of Z, or am I wrong??

no, it's not a subring of Z.

can you give me one more hint please? I am really trying to find out why it is so in any ring, but I´m afraid I´m not able to do it now. I have passed an exam (not algebra) few hours ago, I´m exhausted, but I really want to do my homework...