a,DK: \(x\ne\pm4;x\ne-5\)
\(B=\dfrac{2x}{x+4}+\dfrac{16x}{x^2-16}\)
\(B=\dfrac{2x}{x+4}+\dfrac{16x}{\left(x-4\right).\left(x+4\right)}\)
\(B=\dfrac{2x.\left(x-4\right)}{\left(x-4\right).\left(x+4\right)}+\dfrac{16x}{\left(x-4\right).\left(x+4\right)}\)
\(B=\dfrac{2x^2-8x+16x}{\left(x-4\right).\left(x+4\right)}\)
\(B=\dfrac{2x^2+4x}{\left(x-4\right).\left(x+4\right)}\)\(=\dfrac{2x.\left(x+4\right)}{\left(x-4\right).\left(x+4\right)}=\dfrac{2x}{x-4}\)
Vay \(B=\dfrac{2x}{x-4}\)
b, Ta co: \(A.B=\dfrac{x-4}{x+5}.\dfrac{2x}{x-4}=\dfrac{2x}{x+5}\)\(=\dfrac{2x+10-10}{x+5}=\dfrac{2.\left(x+5\right)}{x+5}-\dfrac{10}{x+5}=2-\dfrac{10}{x+5}\)
De A.B la so nguyen \(\Leftrightarrow2-\dfrac{10}{x+5}\in Z\)
\(\Leftrightarrow\dfrac{10}{x+5}\in Z\)
\(\Leftrightarrow10⋮x+5\)
\(\Leftrightarrow x+5\in\left\{\pm1;\pm2;\pm5\pm10\right\}\)
\(\Leftrightarrow x\in\left\{-6;-3;-7;0;-10;-15\right\}\) do \(x\ne\left\{\pm4;5\right\}\)
Vay x \(\in\left\{-6;-3;-7;0;-10;-15\right\}\)
