First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightPartial Fractionschevron_rightExercise 5.1

Solution

Let:

\frac{3x^2+4x-5}{(x-1)^3}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{(x-1)^3}

Multiply by $(x-1)^3$ :

3x^2+4x-5=A(x-1)^2+B(x-1)+C

Solving gives:

A=3,\quad B=10,\quad C=2

Hence:

\boxed{\frac{3x^2+4x-5}{(x-1)^3}=\frac{3}{x-1}+\frac{10}{(x-1)^2}+\frac{2}{(x-1)^3}}