Bài 13/
a/ ĐKXĐ: \(x\ne ±2\)
b/ \(P=(\dfrac{x}{x^2-4}+\dfrac{2}{2-x}+\dfrac{1}{x+2}):\dfrac{1}{x+2}\\=(\dfrac{x}{(x-2)(x+2)}-\dfrac{2(x+2)}{(x+2)(x-2)}+\dfrac{x-2}{(x+2)(x-2)}).(x+2)\\=\dfrac{x-2(x+2)+(x-2)}{(x+2)(x-2)}.(x+2)\\=\dfrac{x-2x-4+x-2}{x-2}\\=\dfrac{-6}{x-2}\)
Bài 14/
a/ ĐKXĐ:x \(x\ne 0;±1\)
b/ \(P=(\dfrac{x^2+1}{2x}-1)(\dfrac{1}{x-1}+\dfrac{1}{x+1})\\=\dfrac{x^2-2x+1}{2x}.(\dfrac{x+1}{(x-1)(x+1)}+\dfrac{x-1}{(x-1)(x+1)}\\=\dfrac{(x-1)^2}{2x}.\dfrac{x+1+x-1}{(x-1)(x+1)}\\=\dfrac{x-1}{2x}.\dfrac{2x}{x+1}\\=\dfrac{x-1}{x+1}\)
c/ \(P=0\\→\dfrac{x-1}{x+1}=0\\→x-1=0\\↔x=1(KTM)\)
Vậy không có giá trị thỏa mãn \(P=0\)
Bài 15/
a/ ĐKXĐ: \(x\ne -1\)
b/ \(P=(1-\dfrac{x^2}{x^2-x+1}):\dfrac{x^2+2x+1}{x^3+1}\\=\dfrac{x^2-x+1-x^2}{x^2-2x+1}.\dfrac{(x+1)(x^2-x+1)}{(x+1)^2}\\=\dfrac{1-x}{x^2-x+1}.\dfrac{x^2-2x+1}{x+1}\\=\dfrac{1-x}{x+1}\)
c/ \(P=2\\→\dfrac{1-x}{x+1}=2\\→2(x+1)=1-x\\↔2x+2=1-x\\↔3x=-1\\↔x=-\dfrac{1}{3}\)
Vậy \(x=-\dfrac{1}{3}\) thì \(P=2\)
d/ \(P∈\Bbb Z\\→\dfrac{1-x}{x+1}\in\Bbb Z\\→\dfrac{-x-1+2}{x+1}∈\Bbb Z\\→-1+\dfrac{2}{x+1}∈\Bbb Z\\→\dfrac{2}{x+1}∈\Bbb Z\\→2\vdots x+1\\→x+1∈Ư(2)=\{±1;±2\}\)
Ta có bảng:
\(\begin{array}{|c|c|c|}\hline x+1&1&-1&2&-2\\\hline x&0&-2&1&-3\\\hline \quad &tm&tm&tm&tm\\\hline \end{array}\)
Vậy \(x∈\{0;-2;1;-3\}\) thì \(P∈\Bbb Z\)
