Zadanie 1.
[tex]\frac{3}{x+5}-\frac{1}{x-5}+\frac{30}{x^2-25}=\frac{3(x-5)}{(x-5)(x+5)}-\frac{x+5}{(x-5)(x+5)}+\frac{30}{(x-5)(x+5)}=\frac{3x-15-x-5+30}{(x-5)(x+5)}=\frac{2x+10}{(x-5)(x+5)}=\frac{2(x+5)}{(x-5)(x+5)}=\frac{2}{x-5}[/tex]
Zadanie 2.
a)
[tex]f(x)=\frac{(x^3-2x^2+x)(x+1)^2}{x^2-1}[/tex]
Dziedzina:
[tex]x^2-1\neq 0\\(x-1)(x+1)\neq 0\\x\neq 1\land x\neq -1\\D=\mathbb{R}-\{-1,1\}[/tex]
Miejsca zerowe:
[tex](x^3-2x^2+x)(x+1)^2=0\\x(x^2-2x+1)(x+1)^2=0\\x(x-1)^2(x+1)^2=0\\x=0\vee x=1\vee x=-1\ \text{ale } x\in D\\\text {ostatecznie }x=0[/tex]
b)
[tex]f(x)=\frac{1-(x+3)^2}{(x+2)^2}[/tex]
Dziedzina:
[tex](x+2)^2\neq0\\x\neq-2\\D=\mathbb{R}-\{-2\}[/tex]
Miejsca zerowe:
[tex]1-(x+3)^2=0\\(1-x-3)(1+x+3)=0\\(-x-2)(x+4)=0\\-x-2=0\vee x+4=0\\x=-2\vee x=-4\ \text{ale }x\in D\\\text{ostatecznie } x=-4[/tex]
