Lời giải:

Để $\frac{-12}{x}$ là số nguyên thì $x\in Ư(-12)$

$\Rightarrow x\in \left\{\pm 1; \pm 2; \pm 3; \pm 4; \pm 6; \pm 12\right\}(1)$

Để $\frac{15}{x-2}$ nguyên thì $x-2\in Ư(15)$

$\Rightarrow x-2\in \left\{\pm 1; \pm 3; \pm 5; \pm 15\right\}$

$\Rightarrow x\in \left\{1; 3; 5; -1; 7; -3; 17; -13\right\}(2)$

Để $\frac{8}{x+1}$ nguyên thì $x+1\in Ư(8)$

$\Rightarrow x+1\in \left\{\pm 1; \pm 2; \pm 4; \pm 8\right\}$

$\Rightarrow x\in \left\{0; -2; 1; -3; 3; -5; 7; -9\right\}(3)$

Vậy để thỏa mãn đồng thời cả 3 ĐK trên thì $x$ phải cùng thuộc 3 tập $(1); (2); (3)$

$\Rightarrow x\in \left\{1; \pm 3\right\}$

ok how are you

\(A=\frac{x-2}{x+2}=\frac{x^2-4x+4}{x^2-4}=\frac{x^2-4-4x+8}{x^2-4}=1+\frac{-4\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}=1-\frac{4}{x+2}\)

Để   \(A\in Z\)  thì  \(\frac{4}{x+2}\in Z\Leftrightarrow x+2\inƯ\left(4\right)\)

\(\Rightarrow x\in\left\{-6;-4;-3;-1;0;2\right\}\)

\(B=\frac{3x-6}{x+6}=\frac{3x+18-24}{x+6}=\frac{3\left(x+6\right)}{x+6}-\frac{24}{x+6}=3-\frac{24}{x+6}\)

Để  \(B\in Z\)  thì  \(\frac{24}{x+6}\in Z\Leftrightarrow x+6\inƯ\left(24\right)\)

\(\Rightarrow x\in\left\{-30;-18;-14;-12;-10;-9;-8;-7;-5;-4;-3;-2;0;2;6;18\right\}\)

\(C=\frac{10-5x}{x-5}=\frac{-\left(5x-25+15\right)}{x-5}=\frac{-5\left(x-5\right)}{x-5}-\frac{15}{x-5}=-5-\frac{15}{x-5}\)

Để  \(C\in Z\)  thì  \(\frac{15}{x-5}\in Z\Leftrightarrow x-5\inƯ\left(15\right)\)

\(\Rightarrow x\in\left\{-10;0;4;6;10;20\right\}\)

\(D=\frac{8x-2}{2-4x}=\frac{-\left(4-8x\right)+2}{2\left(1-2x\right)}=\frac{-4\left(1-2x\right)}{2\left(1-2x\right)}+\frac{2}{2\left(1-2x\right)}=-2+\frac{1}{1-2x}\)

Để  \(D\in Z\)  thì  \(\frac{1}{1-2x}\in Z\Leftrightarrow1-2x\inƯ\left(1\right)\)

\(\Rightarrow x=0\)

cj kia đúng đó

tham khảo :

x=2 ạ

Để phân số này là số nguyên thì \(2x-5\in\left\{1;-1;19;-19\right\}\)

hay \(x\in\left\{3;2;12;-7\right\}\)

\(\Rightarrow\dfrac{3x^2-6x+5x-10+14}{x-2}=3x+5+\dfrac{14}{x-2}\in Z\\ \Rightarrow x-2\inƯ\left(14\right)=\left\{-14;-7;-2;-1;1;2;7;14\right\}\\ \Rightarrow x\in\left\{-12;-5;0;1;3;4;9;16\right\}\)

Để phân số này là số nguyên thì \(x-2\in\left\{1;-1;2;-2;7;-7;14;-14\right\}\)

hay \(x\in\left\{3;1;4;0;9;-5;16;-12\right\}\)

`( 3x + 2 )/( x + 2 )` nguyên `.`

`=> 3x + 2` \(\vdots\) `x+2`

`=> 3x + 6 - 4` \(\vdots\) `x+2`

`=> 3( x + 2 )-4` \(\vdots\) `x+2`

Do `3( x + 2 )` \(\vdots\) `x+2` mà để `3( x + 2 )-4` \(\vdots\) `x+2`

`=> -4` \(\vdots \)  `x+2` hay `x+2 in Ư_(4) = { +-1 ; +-2 ; +-4 }`

Do `x in ZZ^-`

`=> x + 2 in ZZ` `; x + 2 < 2` 

`=> x + 2 in { +-1 ; -2 ; -4 }`

`=> x in { -1 ; -3 ; -4 ; -6 }`

Vậy `x in { -1;-3;-4;-6}` 

Để \(A=\frac{2x^2+3x+3}{2x+1}\)nguyên thì :

\(\left(2x^2+3x+3\right)⋮\left(2x+1\right)\)

\(\left(2x^2+x+2x+1+2\right)⋮\left(2x+1\right)\)

\(\left[x\left(2x+1\right)+\left(2x+1\right)+2\right]⋮\left(2x+1\right)\)

\(\left[\left(2x+1\right)\left(x+1\right)+2\right]⋮\left(2x+1\right)\)

Vì \(\left(2x+1\right)\left(x+1\right)⋮\left(2x+1\right)\)

\(\Rightarrow2⋮\left(2x+1\right)\)

\(\Rightarrow2x+1\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\)

\(\Rightarrow x\in\left\{0;-1;0,5;-1,5\right\}\)

Vậy....

Bài 8:

\(F=x^2-2x+1+x^2-6x+9=2x^2-8x+10\\ F=2\left(x^2-4x+4\right)+2=2\left(x-2\right)^2+2\ge2\\ F_{min}=2\Leftrightarrow x=2\)

Bài 9:

\(A=-x^2+2x-1+5=-\left(x-1\right)^2+5\le5\\ A_{max}=5\Leftrightarrow x=1\\ B=-x^2+10x-25+2=-\left(x-5\right)^2+2\le2\\ B_{max}=2\Leftrightarrow x=5\\ C=-x^2+6x-9+9=-\left(x-3\right)^2+9\le9\\ C_{max}=9\Leftrightarrow x=3\)