I would appreciate if you can solve this problem. thanks.
What is the greatest integer value of x such that (X^2 + 2X + 5)/(x - 3) is an integer?
hmmmm ok so lets see if i got this right...
In order for (X^2 + 2X + 5)/(x - 3) to be an interger, that means when you do the long division, there is no remainder. Now if you actually try doing the long division of polynomials (here I am assuming you know how to divide polynomials) you are going to notice that there is a remainder of 20/(x-3) now in order for it to be an integer answer, the remainder has to be an integer. The highest number that x can be that would make the remainder an integer would be 23. I think this should work...
As $\displaystyle \frac{x^2 + 2x + 5}{x - 3} = x+5 +\frac{20}{x-3}
$
Then by inspection $\displaystyle x+5 +\frac{20}{x-3}$ the first part of the expression $\displaystyle x+5$ will always give an integer for any integer $\displaystyle x$ so we just have use the same idea to inspect $\displaystyle \frac{20}{x-3}$
As we are looking for an integer there are only so many numbers that will work, I can think of $\displaystyle x = 13,23$