Question

Tìm các số A,B,C để có:

a)\(\dfrac{x^2-x+2}{\left(x-1\right)^3}=\dfrac{A}{\left(x-1\right)^3}+\dfrac{B}{\left(x-1\right)^2}+\dfrac{C}{x-1}\)

b)\(\dfrac{x^2+2x-1}{\left(x-1\right)\left(x^2+1\right)}=\dfrac{A}{x-1}+\dfrac{Bx+C}{x^2+1}\)

Answer 1

a) PT \(\Leftrightarrow\dfrac{x^2-x+2}{\left(x-1\right)^3}=\dfrac{A+B\left(x-1\right)+C\left(x-1\right)^2}{\left(x-1\right)^3}\)

\(\Leftrightarrow x^2-x+2=A+Bx-B+Cx^2-2Cx+C\)

\(\Leftrightarrow x^2-x+2=Cx^2+x\left(B-2C\right)+\left(A+C-B\right)\)

\(\Leftrightarrow\left\{{}\begin{matrix}C=1\\B-2C=-1\\A+C-B=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}A=2\\B=1\\C=1\end{matrix}\right.\)

Answer 2

b: \(\Leftrightarrow\dfrac{x^2+2x-1}{\left(x-1\right)\left(x^2+1\right)}=\dfrac{A\cdot x^2+A+\left(Bx+C\right)\left(x-1\right)}{\left(x^2+1\right)\left(x-1\right)}\)

\(\Leftrightarrow x^2\cdot A+A+x^2\cdot B-x\cdot B+x\cdot C-C=x^2+2x-1\)

\(\Leftrightarrow x^2\left(A+B\right)+x\left(-B+C\right)+A-C=x^2+2x-1\)

=>A+B=1; -B+C=2; A-C=-1

=>A+C=3; A-C=-1; A+B=1

=>A=1; C=2; B=1-A=0