\(A=\left(\dfrac{1-a^3}{a-a^2}+1\right)\left(\dfrac{1+a^3}{1+a}-a\right):\dfrac{\left(1-a^2\right)^3}{1+a}\)(đk: \(a\ne0;a\ne1;a\ne-1\))
\(=\left[\dfrac{\left(1-a\right)\left(1+a+a^2\right)}{a\left(1-a\right)}+1\right]\left[\dfrac{\left(1+a\right)\left(1-a+a^2\right)}{1+a}-a\right]:\dfrac{\left[\left(1-a\right)\left(1+a\right)\right]^3}{1+a}\)
\(=\left(\dfrac{1+a+a^2}{a}+\dfrac{a}{a}\right)\left(1-2a+a^2\right).\dfrac{1+a}{\left(1-a\right)^3\left(1+a\right)^3}\)
\(=\dfrac{1+2a+a^2}{a}.\left(1-a\right)^2.\dfrac{1}{\left(1-a\right)^3\left(1+a\right)^2}\)
\(=\dfrac{\left(1+a\right)^2\left(1-a\right)^2}{a\left(1-a\right)^3\left(1+a^2\right)}\)\(=\dfrac{1}{a\left(1-a\right)}\)
b) \(A>A^2\Leftrightarrow A\left(1-A\right)>0\) \(\Leftrightarrow0< A< 1\)
\(\Leftrightarrow0< \dfrac{1}{a\left(1-a\right)}< 1\)
Tại \(\dfrac{1}{a\left(1-a\right)}>0\) \(\Leftrightarrow a\left(1-a\right)< 0\) \(\Leftrightarrow\left[{}\begin{matrix}a< 0\\a>1\end{matrix}\right.\) (1)
Tại \(\dfrac{1}{a\left(1-a\right)}< 1\) \(\Leftrightarrow\dfrac{1}{a\left(1-a\right)}-1< 0\) \(\Leftrightarrow\dfrac{1-a\left(1-a\right)}{a\left(1-a\right)}< 0\)
\(\Leftrightarrow\dfrac{1-a+a^2}{a\left(1-a\right)}< 0\) \(\Leftrightarrow a\left(1-a\right)< 0\) (vì \(1-a+a^2=\left(a-\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\forall a\))
\(\Leftrightarrow\left[{}\begin{matrix}a< 0\\a>1\end{matrix}\right.\) (2)
Từ (1) (2) ;kết hợp với đk=> \(\left[{}\begin{matrix}a< 0,a\ne-1\\a>1\end{matrix}\right.\)
