Zadanie 1
a) P(x)-(Q(x) + R(x))=[latex]-2x+1-(3 x^{2} -x+2+ x^{3}-4 )=-2x+2-3 x^{2} +x- x^{3}+2= [/latex]
=[latex]- x^{3}-3 x^{2} -x+3 [/latex]
b) P(x)*R(x)-Q(x)=[latex](-2 x^{2} +1)( x^{3}-4 )-(3 x^{2} -x+2)=[/latex]
[latex]=-2 x^{4}+8x+ x^{3}-4-3 x^{2} +x-2 =-2 x^{4}+ x^{3}-3 x^{2} +9x-6 [/latex]
Zadanie 2
a) [latex]6 x^{3} -3 x^{2} +10x-5=(6 x^{3}-3 x^{2})+(10x-5)=[/latex]
[latex]=3 x^{2} (2x-1)+5(2x-1)=(2x-1)(3 x^{2} +5)[/latex]
b) [latex] x^{4}-64=( x^{2})^{2}- 8^{2}=( x^{2} -8)( x^{2} +8) [/latex]
c) [latex]25 x^{2} -20x+4=(5x-2)^{2} [/latex]
Zadanie 3
[latex] frac{ x^{3}-3 x^{2} +x-3 }{x-3}= frac{( x^{3}-3 x^{2} )+(x-3)}{x-3}= frac{ x^{2} (x-3)+(x-3)}{x-3}= frac{(x-3)( x^{2} +1)}{x-3}= x^{2} +1 [/latex]
Zadanie 4
[latex]a) sqrt{(x-3)( x^{2} -4)} [/latex]
[latex](x-3)( x^{2} -4) geq 0[/latex]
[latex]x-3=0[/latex] ∨ [latex] x^{2} -4=0[/latex]
[latex]x=3[/latex] [latex] x^{2} =4[/latex]
[latex]x=2[/latex] ∨ [latex]x=-2[/latex]
x∈ [latex]<-2,2> [/latex] ∨ [latex]<3,+[/latex]∞[latex])[/latex]
Dziedzina [latex]<-2,2> [/latex] ∨ [latex]<3,+[/latex]∞[latex])[/latex]
b) [latex] frac{4 x^{3}-3 x^{5}+3 }{2x(3x-1)} [/latex]
[latex]2x(3x-1)=0[/latex]
[latex]2x=0[/latex] ∨ [latex]3x-1=0[/latex]
[latex]x=0[/latex] [latex]3x=1[/latex]
[latex]x= frac{1}{3} [/latex]
D=R/{0,[latex] frac{1}{3} [/latex]}
Zadanie 5
[latex]b) frac{2x}{2x+2}- frac{x-1}{x+1}=0 [/latex]
Zał.
[latex]2x+2
eq 0[/latex] ∨ [latex]x+1
eq 0[/latex]
[latex]2x
eq -2[/latex] [latex]x
eq -1[/latex]
[latex]x
eq -1[/latex]
[latex] frac{2x}{2(x+1)} - frac{x-1}{x+1}=0 [/latex]
[latex] frac{x-x+1}{x+1}=0 [/latex]
[latex] frac{1}{x+1}=0 [/latex]
[latex]1
eq 0[/latex]
sprzeczność
Jak coś nie jasne to pytaj :)
