1. a) [latex]-3x^4+3x^3+6x^2=-3x^2(x^2-x-2)=-3x^2(x+1)(x-2)[/latex] [latex]x^2-x-2[/latex] [latex]Delta=(-1)^2-4cdot (1)cdot (-2)=1+8=9[/latex] [latex]sqrt{Delta}= sqrt{9}=3[/latex] [latex]x_1= frac{1-3}{2}=-1[/latex] [latex]x_2= frac{1+3}{2}=2[/latex] b) [latex]x^4-625=(x^2-25)(x^2+25)=(x-5)(x+5)(x^2+25)[/latex] c) [latex]x^3-x=x(x^2-1)=x(x-1)(x+1)[/latex] d) [latex]4x^2+4x+1=(2x+1)^2[/latex] e) [latex](x-3)(2x-1)-(4x-2)(x-3)=(x-3)(2x-1-4x+2)=(x-3)(-2x+1)[/latex] f) [latex]x^3- frac{1}{8}=x^3-( frac{1}{2} )^3=(x- frac{1}{2})(x^2+ frac{1}{2}x+ frac{1}{4})[/latex] g) [latex]27x^3+64=(3x)^3+4^3=(3x + 4)(9x^2-12x + 16)[/latex] ============================== 2. [latex]3x^3-6x^2+4x-8=0[/latex] [latex]3x^2(x-2)+4(x-2)=0[/latex] [latex](x-2)(3x^2+4)=0[/latex] [latex]x-2=0[/latex] [latex]x=2[/latex] ___________ [latex]x^3=4x[/latex] [latex]x^3-4x=0[/latex] [latex]x(x^2-4)=0[/latex] [latex]x(x-2)(x+2)=0[/latex] [latex]x=0 lub x-2=0 lub x+2=0[/latex] [latex]. x=2 lub x=-2[/latex] ___________ [latex]x^3+8=0[/latex] [latex](x + 2)(x^2 - 2x + 4)=0[/latex] [latex]x+2=0[/latex] [latex]x=-2[/latex] ___________ [latex]x^4-16=0[/latex] [latex](x^2-4)(x^2+4)=0[/latex] [latex](x-2)(x+2)(x^2+4)=0[/latex] [latex]x-2=0 lub x+2=0[/latex] [latex]x=2 lub x=-2[/latex] ============================== 3. [latex](x-4)(x+1)>0[/latex] rysujemy oś i odczytujemy [latex]xin(- infty ,-1)cup (4,+ infty )[/latex] _________________ [latex]x^2 ge 3x[/latex] [latex]x^2-3x ge 0[/latex] [latex]x(x-3) ge 0[/latex] rysujemy oś i odczytujemy [latex]xin(- infty ,0>cup <3,+ infty )[/latex] _________________ [latex]9-x^2>0[/latex] [latex](3-x)(3+x)>0[/latex] rysujemy oś i odczytujemy [latex]xin(-3,3)[/latex] _________________ [latex](x+1)(x-4)-(x+1)(x+6) ge 0[/latex] [latex](x+1)(x-4-x-6) ge 0[/latex] [latex]-10(x+1) ge 0 /:(-10)[/latex] [latex]x+1 le 0[/latex] [latex]x le -1[/latex] [latex]xin(- infty ,-1>[/latex]
