Need help with my first real proof!

[jgurley]

Hi, I am doing my first real proof and am completely stuck!!
I have proved the conclusion by using definition and correct notation. My problem is that in my mind I know that it's true and why, but I can't seem to connect the definition to the hypothesis correctly. Here is the problem and what I have so far:
If an integer m is a root of the polynomial x^3 + kx^2 + jx + i, where k, j, and i are integers, then m divides i.

A: Interger m is a root of x^3 + kx^2 +jx + i, where k, j, and i are inegers
A1: where x=m, m^3 + km^2 + jm + i = 0, by definition of a root

B: m divides i
B1: definition--> a divides b if b=ca, for some integer c.
B2: definition ----> proposition
a ----> m
b ----> i
c ----> c
B3: by definition, i = cm, and by manipulation of this equation, i/m = c

I have shown that m divides i but can't figure out how to connect A and B any further, I have tried replacing different parts fo the formula but j and k are giving me issues to show that A and B are connected! Where do I go from here?? I know this is true but can't figure out how to prove this!! Please help!!!

 

[daon2]

Manipulate the equation further to get: m(-m^2-km-j) = i. Let c = (-m^2-km-j) is an integer. Then (m)(c)=i, that is, m divides i.

 

[jgurley]

Thank you so much...I did all of that but because I didn't thoroughly read that m was also an integer in the proposition, I failed to place that -m^2-km-j would also be an integer that could be called c! I keep thinking I was making this way more complicated than it was, and it was all due to miss reading the original proposition! Thank you!!!!

 

[pka]

If an integer m is a root of the polynomial x^3 + kx^2 + jx + i, where k, j, and i are integers, then m divides i

As reply #3 points out \(\displaystyle m\ne 0\).
The easy way to prove this is to recall that \(\displaystyle i\) is the product of the roots. So it is divisible by \(\displaystyle m\).