b) [latex]10x^5+30x^4=10x^4(x+3) o D=Rsetminus { -3;0} \ 15x^7-20x^5=5x^5(3x^2-4) \ frac{15x^7-20x^5}{10x^5+30x^4}=frac{5x^5(3x^2-4)}{10x^4(x+3)}=frac{x(3x^2-4)}{2(x+3)}[/latex] c) [latex]6x^4-2x^2=0 \ 2x^2(3x^2-1)=0 \ 2x^2=0 vee 3x^2-1=0 \ x=0 vee 3x^2=1 \ x=0 vee x^2=frac13 \ x=0 vee x=-sqrt{frac13} vee x=sqrt{frac13} \ D=Rsetminus{-sqrtfrac13; 0;sqrtfrac13}\ frac{3x^3-x}{6x^4-2x^2}=frac{x(3x^2-1)}{2x^2(3x^2-1)}=frac1{2x}[/latex] f) [latex]9-3x=0 \ 9=3x \ 3=x \ D=Rsetminus{3} \ frac{6x^2-18x}{9-3x}=frac{6x(x-3)}{3(3-x)}=frac{6x(x-3)}{-3(x-3)}=frac{6x}{-3}=-2x[/latex] h) [latex]16x^2-100=0 |:4 \ 4x^2-25-0 \ 4x^2=25 \ x^2=frac{25}4 \ x=-sqrtfrac{25}4 vee x=sqrtfrac{25}4 \ x=-frac52 vee x=frac52 \ D=Rsetminus{-frac52;frac52} \ frac{20x-50}{16x^2-100}=frac{10(2x-5)}{4(4x^2-25)}=frac{5(2x-5)}{2(4x^2-25)}=frac{5(2x-5)}{2(2x-5)(2x+5)}=frac5{2(2x+5)}[/latex] pod koniec przykładu h) wykorzystałem wzór [latex]a^2-b^2=(a-b)(a+b)[/latex] i) [latex]x^2-4=0 \ x^2-2^2=0 \ (x-2)(x+2)=0 \ x-2=0 vee x+2=0 \ x=2 vee x=-2 \ D=Rsetminus{-2;2} \ frac{x^3-8}{x^2-4}=frac{x^2-2^3}{x^2-2^2}=frac{(x-2)(x^2+2x+4)}{(x-2)(x+2)}=frac{x^2+2x+4}{x+2}[/latex] wykorzystałem tutaj wzór [latex]a^3-b^3=(a-b)(a^2+acdot b+b^2)[/latex] j) [latex]1000x^3+1=0 \ 1000x^3=-1 \ x^3=-frac1{1000} \ x= sqrt[3]{-frac1{1000}} \ x=-frac1{10} \ D=Rsetminus{-frac1{10}} \ frac{300x^2-3}{1000x^3+1}=frac{3(100x^2-1)}{(10x)^3+1^3}=frac{3[(10x)^2-1^2]}{(10x+1)[(10x^2-10xcdot 1+1^2]}=\=frac{3(10x-1)(10x+1)}{(10x+1)(100x^2-10x+1)}=frac{3(10x-1)}{100x^2-10x+1}[/latex] k) [latex]x^2-6x-7=0 \ x^2+x-x-6x-7=0 \ x^2+x-7x-7=0 \ x(x+1)-7(x+1)=0 \ (x-7)(x+1)=0 \ x-7=0 vee x+1=0 \ x=7 vee x=-1 \ D=Rsetminus{-1;7} \ frac{x^2-5x-14}{x^2-6x-7}=frac{x^2-7x+2x-14}{(x-7)(x+1)}=frac{x(x-7)+2(x-7)}{(x-7)(x+1)}=frac{(x+2)(x-7)}{(x-7)(x+1)}=frac{x+2}{x+1}[/latex] l) [latex]x^2+6x+5=0 \ x^2+x+5x+5=0 \ x(x+1)+5(x+1)=0 \ (x+5)(x+1)=0 \ x+5=0 vee x+1=0 \ x=-5 vee x=-1 \ D=Rsetminus{-5;-1} \ frac{x^2-3x-4}{x^2+6x+5}=frac{x^2+x-4x-4}{(x+5)(x+1)}=frac{x(x+1)-4(x+1)}{(x+5)(x+1)}=frac{(x-4)(x+1)}{(x+5)(x+1)}=frac{x-4}{x+5}[/latex]
b] mianownik; 10x do potegi 5+30x do potegi 4=10x do potegi 4(x+3) 10 x do potegi 4(x+3)≠0 D: x≠0 i x≠ -3 licznik: 15x do potegi 7-20x do potegi 5=5x do potegi 5(3x²-4) całosc=x(3x²-4) / 2(x+3)=(3x³-4x) / (2x+6) c] mianownik=2x²(3x²-1) D=x≠0 i x≠√3/3 i x≠-√3/3 licznik=x(3x²-1) całosc=x/2x²=1/ 2x f] 9-3x≠0 3x≠9 x≠3=D licznik=6x(x-3) mianownik=-3(x-3) całosc=6x/-3=-2x h] mianownik=16x²=100 x²=100/16 x²=25/4 D= x≠5/2 i x≠-5/2 mianownik=(4x+10_(4x-10) licznik=5(4x-10) całosc=5/(5x+10) i] mianownik=(x+2)(x-2) D=x≠2 i x≠-2 licznik=x³-8=(x-2)(x²+2x+4) całosc=(x²+2x+4)/(x+2) j] mianownik=(10x+1)(100x²-10x+1) D; 1000x³≠-1 x³≠-1/1000 x≠-1/10 licznik; =3(100x²-1)=3(10x+1)(10x-1) całosc= 3(10x-1) / (100x²-10x+1) k] mianownik; Δ=36+28=64 x1=[6-8]/2=-1 x2=[6+8]/2=7 D: x≠-1 i x≠7 mianownik=(x+1)(x-7) licznik=Δ=25+56=81 x1=[5-9]/2=-2 x2=[5+9]/2=7 licznik=(x+2)(x-7) całosc=(x+1)/(x-7) l] mianownik" Δ=36-20=16 D; x1≠[-6-4]/2=-5 i x2≠ [ -6+4]/2=-1 mianownik=(x+5)(x+1) licznik; Δ=9+16=25 x1=[3+5]/2=4 x2=[3-5]/2=-1 licznik=(x-4)(x+1) caosc=(x-4)/(x+5)
