Question

Partial fraction decomposition: 1/(x³ - 11x² + 24x + 36)

Answer 1

\[x^3-11x^2+24x+36=x^3+x^2-12x^2-12x+36x+36=\]\[=x^2(x+1)-12x(x+1)+36(x+1)=\]\[=(x+1)(x^2-12x+36)=(x+1)(x-6)^2\]\[\frac{1}{x^3-11x^2+24x+36}=\frac{1}{(x+1)(x-6)^2}=\]\[=\frac{A}{x+1}+\frac{B}{x-6}+\frac{Cx+D}{(x-6)^2}\quad/\cdot (x+1)(x-6)^2\]\[1=A(x-6)^2+B(x+1)(x-6)+(Cx+D)(x+1)\]\[A=\frac{1}{49},\quad B=-\frac{1}{49},\quad C=0,\quad D=\frac{1}{7}\]

Answer 2

1/[49(x+1)]-1/[49(x-6)]+1/[7(x-6)^2]