Question

xác định a,b,c sao cho :

\(\dfrac{1}{x^3-1}=\dfrac{a}{x-1}+\dfrac{bx+c}{x^2+x+1}\)

Answer 1

\(\Leftrightarrow\frac{1}{x^3-1}=\frac{a\left(x^2+x+1\right)+\left(bx+c\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)

\(\Leftrightarrow\frac{1}{x^3-1}=\frac{ax^2+ax+a+bx^2+cx-bx-c}{x^3-1}\)

\(\Leftrightarrow\left(a+b\right)x^2+\left(a-b+c\right)x+a-c=1\)

\(\Leftrightarrow\left\{{}\begin{matrix}a+b=0\\a-b+c=0\\a-c=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a+b=0\\2a+c=0\\a-c=1\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}3a=1\\a+b=0\\a-c=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=\frac{1}{3}\\b=-\frac{1}{3}\\c=-\frac{2}{3}\end{matrix}\right.\)