3 działania

[zaq1a]

a) \(\frac{x - 2}{x - 3} - \frac{x - 1}{x + 3}\)

b) \(\frac{x - 1}{3x - 6} + \frac{x + 1}{3x + 6}\)

c) \(\frac{x^4 - 1}{2x^2 + 2} * \frac{4}{3x^2 - 3}\)

[irena]

a)

Zastrzeżenia
\(x \in R \setminus \left\{-3;\ 3 \right\}\)

\(\frac{x-2}{x-3}-\frac{x-1}{x+3}=\frac{(x-2)(x+3)-(x-1)(x-3)}{(x-3)(x+3)}=\frac{x^2+3x-2x-6-x^2-3x+x-3}{x^2-9}=\frac{-x-9}{x^2-9}=\frac{x+9}{9-x^2}\)

[irena]

b)

Zastrzeżenia:
\(x \in R \setminus \left\{-2;\ 2 \right\}\)


\(\frac{x-1}{3(x-2)}+\frac{x+1}{3(x+2)}=\frac{(x-1)(x+2)+(x+1)(x-2)}{3(x-2)(x+2)}=\frac{x^2+2x-x-2+x^2-2x+x-2}{3(x^2-4)}=\frac{2x^2-4}{3x^2-12}\)

[irena]

c)
\(x^4-1=(x^2-1)(x^2+1)=(x-1)(x+1)(x^2+1)\\1x^2+2=2(x^2+1)\\3x^2-3=3(x^2-1)=3(x-1)(x+1)\)

Zastrzeżenia:
\(x \in R \setminus \left\{-1;\ 1 \right\}\)

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