F. rozłożyć na ułamki proste.

[faraus]

\(\displaystyle{ \frac{x ^{2} }{x ^{3}+ x ^{2} - 4x -4 }}\) . Rozłożyć na rzeczywiste ułamki proste.

[msx100]

\(\displaystyle{ W_1(x) = \frac{x^2}{x^3+x^2-4x-4} =\frac{x^2}{(x^3 - 4x) + (x^2-4)} = \frac{x^2}{x(x^2-4) + (x^2-4)} = \frac{x^2}{(x^2-4)(x+1)} = \frac{x^2}{(x-2)(x+2)(x+1)} \equiv \frac{A}{x-2} + \frac{B}{x+2} + \frac{C}{x+1} = W_2(x) \\
\frac{A}{x-2} + \frac{B}{x+2} + \frac{C}{x+1} = \frac{A(x+2)(x+1) + (x-2)B(x+1) + (x-2)(x+2)C}{(x-2)(x+2)(x+1)} = \frac{x^2(A+B+C) + x(3A - B) + (2A - 2B -4C)}{(x-2)(x+2)(x+1)}}\)
Aby wieloman \(\displaystyle{ W_1}\) byl rowny wielomianowi \(\displaystyle{ W_2}\) musi zachodzic:
\(\displaystyle{ \left\{\begin{array}{l} A+B+C = 1\\3A -B = 0\\2A-2B-4C = 0 \end{array} ft\{\begin{array}{l} A = \frac{1}{3} \\B = 1 \\C = - \frac{1}{3} \end{array}}\)
\(\displaystyle{ \frac{x^2}{x^3+x^2-4x-4} = \frac{1}{3(x-2)} + \frac{1}{x+2} - \frac{1}{3(x+1)}}\)