[latex]a) \\x^3-6x^2+11x-6=0 \\x^3-3x^2-3x^2+9x+2x-6=0 \\x^2( x -3)-3x(x -3)+2(x-3)=0 \ \(x-3)(x^2-3x+2) =0[/latex] [latex](x-3)(x^2- x-2x+2) =0\\(x-3)[x(x - 1)-2(x-1 )] =0\ \(x-3)(x-1)(x-2)=0\\x-3=0 lub x-1=0 lub x-2=0\\x=3 lub x =1 lub x =2[/latex] [latex]b)\\ x^5-7x^4+12x^3=0\\x^{3}(x^2-7x +12 )=0 \\ x^{3}(x^2-3x-4x +12 )=0 \\ x^{3}[(x^2-3x-4x +12 )]=0[/latex] [latex]x^3 [x(x -3)-4(x -3) ] =0\\x^3=0 lub x-3=0 lub x -4 =0\\ x =0 lub x=3 lub x =4[/latex] [latex]c)\\ (x^2-1)(7-x)(1-x) leq 0 \\(x -1)(x+1)(7-x)(1-x) leq 0 \ \miejsca zerowe :\\x -1=0 i x+1=0 i 7-x=0 i 1-x =0[/latex] [latex]x =1 i x =-1 i x=7 i x =1 \ \ x in (-1,1)cup (1,7)[/latex] [latex]d)\\ (x+1)(x ^2-5x+6)>0\\(x+1)(x ^2-3x-2x+6)>0\\(x+1)[x(x -3 )+2(x-3) ]>0\\(x+1) (x -3 ) (x+2) >0[/latex] [latex]miejsca zerowe : \\ x+1=0 i x -3=0 i x+2=0\\x=-1 i x=3 i x=-2 \\x in(-1,-2)cup (3,+infty )[/latex] [latex]e)\\ x^4=2x^6+x^5\\ 2x^6+x^5-x^4=0\\x^4(2x^2+x -1)=0\\x^4(2x^2+2x-x -1)=0[/latex] [latex]x^4[2x( x+1)-(x +1)]=0\\x^4(x+1)(2x-1)=0\\x^4=0 lub x+1=0 2x-1 =0\\x =0 lub x=-1 2x=1\\x =0 lub x=-1 x=frac{1}{2}\\[/latex] [latex]f) \\x^3-6x geq x-6\\x^3-6x -x+6geq 0\\ x^3-7x +6geq 0\\x^3+3x^2-3x^2-9x +2x+6geq 0[/latex] [latex]x^2(x +3 )-3x(x+3)+2(x +3)geq 0 \\(x+3)( x^2 -3x +2 )geq 0\\(x+3)( x^2 - x-2x +2 )geq 0\\(x+3)[x( x-1 )-2( x -1 )]geq 0\\(x+3) ( x-1 ) ( x -2 ) geq 0[/latex] [latex]miejsca zerowe : \ \ x+3=0 i x-1=0 i x -2 =0 \ \x=-3 i x=1 i x =2 \\x in < -3 , 1> cup <2,+infty )[/latex]
