What are the possible values of m and n for which Q divides P?

[larry91]

Hi everybody! I have this problem: Either P = (X+2)^m+(X+3)ⁿ and Q = x²+5x+7;
Determine m, n such that Q | P;( m, n = ? (Q divide P));

May you help me please?
Thank You!

 

[micromass]

We'll be happy to help you, but not unless you show some effort first...

 

[larry91]

Ok! I try this, but I dislike this method:
Either 'P mod Q' the rest of the P at Q rapport;
(x+2)² mod Q = -(x+3)
(x+2)³ mod Q = (x+2)²(x+2) mod Q = (-(x+3)(x+2)) mod Q = 1
=> the debris of (x+2)^m at Q are repeated from 3 to 3;
(x+3)² mod Q = x+2
(x+3)³ mod Q = ((x+2)(x+3)) mod Q = -1
(x+3)⁴ mod Q = -(x+3) mod Q = -(x+3)
(x+3)⁵ mod Q = -(x+2)
(x+3)⁶ mod Q = 1
=> the debris of (x+3)ⁿ at Q are repeated from 6 to 6;

P mod Q = ((x+2)^m mod Q + (x+3)ⁿ mod Q) mod Q = 0;

1) If m mod 3 = 0 => (x+2)^m mod Q = 1 => (x+3)ⁿ mod Q = -1 => n mod 6 = 3 => m = 3*u; n = 6*v+3

2) If m mod 3 = 1 => (x+2)^m mod Q = x+2 => (x+3)ⁿ mod Q = -(x+2) => n mod 6 = 5 => m = 3*u+1; n = 6*v+5

3) If m mod 3 = 2 => (x+2)^m mod Q = -(x+3) => (x+3)ⁿ mod Q = x+3 => n mod 6 = 1 => m = 3*u+2; n = 6*v+1

=> S={{m = 3*u; n = 6*v+3}, {m = 3*u+1; n = 6*v+5}, {m = 3*u+2; n = 6*v+1}}