a)
x^5-2x^4-3x^3+27x^2-54x-81=0 …-54x=-81x+27x
x^5-2x^4-3x^3+27x^2-81x+27x-81=0
x^4(x-3)+x^3(x-3)+27x(x-3)+27(x-3)=0
(x-3)(x^4+x^3+27x+27)=0
(x-3)[x^3(x+1)+27(x+1)]=0
(x-3)(x+1)(x^3+27)=0
(x-3)(x+1)(x^3+3^3)=0 suma sześcianów - wzór a^3+b^3=(a+b)(a^2-ab+b^2)
(x-3)(x+1)(x+3)(x^2-3x+9)=0
x-3=0\vee x+1=0\vee x+3=0\vee x^2-3x+9=0
x=3\vee x=-1\vee x=-3
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x^2-3x+9=0
a=1, b=-3, c=9
\Delta=b^2-4ac=9-4*9=-27
\Delta <0 brak pierwiastków
b)
x^6-2x^5+3x^4-4x^2+8x-12=0
x^4(x^2-2x+3)-4(x^2+2x-4)=0
(x^2-2x+3)(x^4-4)=0
(x^2-2x+3)(x^2-2)(x^2+2)=0
(x^2-2x+3)(x-\sqrt2)(x+\sqrt2)(x^2+2)=0
x^2-2x+3=0\vee x-\sqrt2=0\vee x+\sqrt2=0 , x^2+2>0 dla każdej liczby rzeczywistej \mathbb R
a=1, b=-2, c=3
\Delta=4-4*1*3=-8 brak pierwiastków
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x-\sqrt2=0
x=\sqrt2
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x+\sqrt2=0
x=-\sqrt2
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x^2+2>0 dla każdej liczby \mathbb R
x=\sqrt2\vee x=-\sqrt2
c)
x^5-x^4+2x^3-2x^2+x-1=0
x^4(x-1)+2x^2(x-1)+(x-1)=0
(x-1)(x^4+2x^2+1)=0
(x-1)(x^2+1)^2=0 …(a+b)^2=a^2+2ab+b^2
x-1=0 , x^2+1>0 dla każdej liczby \mathbb R
x=1
d)
x^5-3x^4+6x^3-18x^2+9x-27=0
x^4(x-3)+6x^2(x-3)+9(x-3)=0
(x-3)(x^4+6x^2+9)=0
(x-3)(x^2+3)^2=0
x-3=0 , x^2+3>0 dla każdej liczby \mathbb R
x=3
e)
x^5+2x^4-3x^3-6x^2+2x+4=0
x^4(x+2)-3x^2(x+2)+2(x+2)=0
(x+2)(x^4-3x^2+2)=0
(x+2)(x^4-2x^2-x^2+2)=0
(x+2)[x^2(x^2-2)-(x^2-2)]=0
(x+2)(x^2-2)(x^2-1)=0
(x+2)(x-\sqrt2)(x+\sqrt2)(x-1)(x+1)=0
x=-2\vee x=\sqrt2\vee x=-\sqrt2\vee x=1\vee x=-1