[latex]a) 3x^3+2x^2-12x-8=0\\x^2(3x+2)-4(3x+2)=0\\(3x+2)(x^2-4)=0iff3x+2=0 vee x^2-4=0\\3x=-2 vee x^2=4\\x=-frac{2}{3} vee x=-2 vee x=2[/latex] ========================================================= [latex]b) 20x^3-12x^2-5x+3=0\\4x^2(5x-3)-1(5x-3)=0\\(5x-3)(4x^2-1)=0iff5x-3=0 vee 4x^2-1=0\\5x-3=0 vee 4x^2=1\\5x=3 vee x^2=frac{1}{4}\\x=frac{3}{5} vee x=-frac{1}{4} vee x=frac{1}{4}[/latex] ========================================================= [latex]c) 6x^3+5x^2+18x+15=0\\x^2(6x+5)+3(6x+5)=0\\(6x+5)(x^2+3)=0iff6x+5=0 vee x^2+3=0\\6x=-5 vee x^2=-3-falsz\\x=-frac{5}{6}[/latex] ========================================================= [latex]d) 10x^3-15x^2+2x-3=0\\5x^2(2x-3)+1(2x-3)=0\\(2x-3)(5x^2+1)=0iff2x-3 vee 5x^2+1=0\\2x=3 vee 5x^2=-1\\x=frac{3}{2} vee x^2=-frac{1}{5}-falsz[/latex] ========================================================= [latex]e) 12x^3+8x^2-75x-50=0\\4x^2(3x+2)-25(3x+2)=0\\(3x+2)(4x^2-25)=0iff3x+2=0 vee 4x^2-25=0\\3x=-2 vee 4x^2=25\\x=-frac{2}{3} vee x^2=frac{25}{4}\\x=-frac{2}{3} vee x=-frac{5}{2} vee x=frac{5}{2}[/latex] ========================================================= [latex]f) 27x^3+45x^2-12x-20=0\\9x^2(3x+5)-4(3x+5)+0\\(3x+5)(9x^2-4)+0iff3x+5=0 vee 9x^2-4=0\\3x=-5 vee 9x^2=4\\x=-frac{5}{3} vee x^2=frac{4}{9}\\x=-frac{5}{3} vee x=-frac{2}{3} vee x=frac{2}{3}[/latex] ========================================================= [latex]g) 10x^3-35x^2+2x-7=0\\5x^2(2x-7)+1(2x-7)=0\\(2x-7)(5x^2+1)=0iff2x-7=0 vee 5x^2+1=0\\2x=7 vee 5x^2=-1\\x=frac{7}{2} vee x^2=-frac{1}{5}-falsz[/latex]
a)3x^3+2x^2-12x-8=x^2(3x+2)-4(3x+2)=(x^2-4)(3x+2)=3(x+2/3)(x-2)(x+2)=0 x= -2/3 lub x=2 lub x= -2 b)20x^3-12x^2-5x+3=4x^2(5x-3)-1(5x-3)=(4x^2-1)(5x-3)=4(x^2-1/4)(5x-3)=20(x-1/2)(x+1/2) (x-3/5)=0 x= 1/2 lub x= -1/2 lub x=3/5 c)6x^3+5x^2+18x+15=x^2(6x+5)+3(6x+5)=(x^2+3)(6x+5)=6(x+5/6)(x^2+3)=0 x^2+3=0 brak rozw. gdyz Δ<0 x+5/6=0 x= -5/6 d)10x^3-15x^2+2x-3=5x^2(2x-3)+1(2x-3)=(5x^2+1)(2x-3)=2(5x^2+1)(x-1,5)=0 5x^2-1=0 brak rozw. Δ<0 x-1,5=0 x=1,5 e)12x^3+8x^2-75x-50=4x^2(3x+2)-25(3x+2)=(4x^2-25)(3x+2)=12(x^2-6,25)(x+2/3)= 12(x-2,5)(x+2,5)(x+2/3)=0 x=2,5 lub x= -2,5 lub x= -2/3 f)27x^3+45x^2-12x-20=9x^2(3x+5)-4(3x+5)=(9x^2-4)(3x+5)=27(x^2-4/9)(x+5/3)=27(x-2/3)(x+2/3)(x+5/3)=0 x=2/3 lub x= -2/3 lub x= -5/3 g)10x^3-35x^2+2x-7=5x^2(2x-7)+1(2x-7)=(5x^2+1)(2x-7)=5(x^2+1/5)(2x-7)=10(x^2+1/5)(x-3,5)=0 x^2+1/5=0 brak rozw. Δ<0 x-3,5=0 x=3,5
