Funkcję wymierną [latex] frac{4 x^{3}-3 x^{2} -2x-3 }{ x^{4}-1 } [/latex] rozłożyć na rzeczywiste ułamki proste.

[latex]frac{4x^3-3x^2-2x-3}{x^4-1}=frac{4x^3-3x^2-2x-3}{(x^2-1)(x^2+1)}=frac{4x^3-3x^2-2x-3}{(x-1)(x+1)(x^2+1)}=\=frac{A}{x-1}+frac{B}{x+1}+frac{Cx+D}{x^2+1}\P=A(x+1)(x^2+1)+B(x-1)(x^2+1)+(Cx+D)(x^2-1)=\=A(x^3+x+x^2+1)+B(x^3+x-x^2-1)+Cx^3-Cx+Dx^2-D=\=Ax^3+Ax^2+Ax+A+Bx^3-Bx^2+Bx-B+Cx^3+Dx^2-Cx-D[/latex] [latex]\=(A+B+C)x^3+(A-B+D)x^2+(A+B-C)x+A-B-D \left{egin{matrix} A &+B &+C &=4 &Rightarrow A=4-B-C \ A &-B &+D &=-3 &\ A &+B &-C &=-2 & \ A&-B &-D &=-3 & end{matrix} ight. \left{egin{matrix} 4 &-B &-C &-B &+D &=-3 & \ 4 &-B &-C &+B &-C &=-2 & \ 4& -B &-C &-B &-D &=-3 & end{matrix} ight. \left{egin{matrix} -2B &-C &+D &=-7 & & & \ & & -2C &=-6 & Rightarrow C=3 & & \ -2B&-C &-D &=-7 & & & end{matrix} ight.[/latex] [latex]\underline{left{egin{matrix} -2B &+D &=-4 & & \ -2B &-D &=-4 & & end{matrix} ight.}+ \-4B quad quad quad =-8 \B=2 \A=4-2-3=-1 \D=-7+3+4=0[/latex] [latex]frac{4x^3-3x^2-2x-3}{x^4-1}=-frac{1}{x-1}+frac{2}{x+1}+frac{3x}{x^2+1}[/latex]