a) \(B=\dfrac{x+2}{2x-4}-\dfrac{x-2}{2x+4}-\dfrac{8}{4-x^2}\)
\(B=\dfrac{x+2}{2\left(x-2\right)}-\dfrac{x-2}{2\left(x+2\right)}-\dfrac{-8}{\left(x-2\right)\left(x+2\right)}\)
\(B=\dfrac{\left(x+2\right)^2-\left(x-2\right)^2+16}{2\left(x+2\right)\left(x-2\right)}\)
\(B=\dfrac{\left(x^2+4x+4\right)-\left(x^2-4x+4\right)+16}{2\left(x+2\right)\left(x-2\right)}\)
\(B=\dfrac{x^2+4x+4-x^2+4x-4+16}{2\left(x+2\right)\left(x-2\right)}\)
\(B=\dfrac{8x+16}{2\left(x+2\right)\left(x-2\right)}\)
\(B=\dfrac{8\left(x+2\right)}{2\left(x+2\right)\left(x-2\right)}\)
\(B=\dfrac{4}{x-2}\)
b) Ta có: \(B=\dfrac{1}{2}\) khi:
\(\dfrac{4}{x-2}=\dfrac{1}{2}\) (ĐK: \(x\ne2\))
\(\Leftrightarrow8=x-2\)
\(\Leftrightarrow x=10\left(tm\right)\)
c) B nguyên khi \(\dfrac{4}{x-2}\) nguyên
Mà: \(\dfrac{4}{x-2}\) nguyên khi 4 ⋮ \(x-2\)
\(\Rightarrow x-2\inƯ\left(4\right)=\left\{-1;1;-2;2;4;-4\right\}\)
Ta có bảng:
| x-2 | -1 | 1 | -4 | 4 | -2 | 2 |
| x | 1 | 3 | -2 | 6 | 0 | 4 |
Vậy B nguyên khi \(x\in\left\{1;3;-2;6;0;4\right\}\)
