Question

Answer 1

1) \(\dfrac{2x-5}{x^2-x-1}=\dfrac{A}{-x^2+x+1}\)

\(\Leftrightarrow\dfrac{5-2x}{-x^2+x+1}=\dfrac{A}{x^2+x+1}\Leftrightarrow A=5-2x\)

2) \(\dfrac{x^3-\dfrac{1}{8}}{x^2+\dfrac{x}{2}+\dfrac{1}{4}}=\dfrac{A}{x}\)

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

\(\Leftrightarrow\dfrac{A}{x}=x-\dfrac{1}{2}\Leftrightarrow A=\left(x-\dfrac{1}{2}\right)x=x^2-\dfrac{1}{2}x\)