First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightPartial Fractionschevron_rightExercise 5.1

Solution

Let:

\frac{4x^2-3x+1}{(x+1)(x-1)^2}=\frac{A}{x+1}+\frac{B}{x-1}+\frac{C}{(x-1)^2}

Multiply by $(x+1)(x-1)^2$ :

4x^2-3x+1=A(x-1)^2+B(x+1)(x-1)+C(x+1)

Solving gives:

A=2,\quad B=2,\quad C=1

Hence:

\boxed{\frac{4x^2-3x+1}{(x+1)(x-1)^2}=\frac{2}{x+1}+\frac{2}{x-1}+\frac{1}{(x-1)^2}}