a)
{\sqrt2x^3+2x^2+2x+\sqrt8=\sqrt2x^3+(\sqrt2)^2*x^2+2x+2\sqrt2=\sqrt2x^2(x+\sqrt2)+2(x+\sqrt2)=}
=(x+\sqrt2)(\sqrt2x^2+2) postać iloczynowa
b)
x^3+x^2-2-\sqrt8=x^3-\sqrt{8}+x^2-2=x^3-(\sqrt2)^3+x^2-(\sqrt2)^2=
=(x-\sqrt2)(x^2+\sqrt2x+2)+(x-\sqrt2)(x+\sqrt2)=
=(x-\sqrt2)(x^2+\sqrt2x+2+x+\sqrt2)=
=(x-\sqrt2)(x^2+\sqrt2x+x+2+\sqrt2) postać iloczynowa
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x^2+\sqrt2x+x+2+\sqrt2=0
x^2+(\sqrt2+1)x+2+\sqrt2=0
a=1 , b=\sqrt2+1 , c=2+\sqrt2
{\Delta=b^2-4ac=(\sqrt2+1)^2-4*1*(2+\sqrt2)=2+2\sqrt2+1-8-4\sqrt2=-5-2\sqrt2<0}
brak pierwiastków
c)
x^3+x^2-2-\sqrt[3]{4}=x^3-2+x^2-\sqrt[3]{4}=x^3-(\sqrt2)^3+x^2-(\sqrt[3]{2})^2=
=(x-\sqrt[3]2)(x^2+\sqrt[3]{2}x+\sqrt[3]{4})+(x-\sqrt[3]2)(x+\sqrt[3]2)=
=(x-\sqrt[3]{2})(x^2+\sqrt[3]{2}x+\sqrt[3]{4}+x+\sqrt[3]{2})=
=(x-\sqrt[3]{2})(x^2+\sqrt[3]{2}x+x+\sqrt[3]{4}+\sqrt[3]{2}) postać iloczynowa
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x^2+(\sqrt[3]2+1)x+\sqrt[3]{4}+\sqrt[3]{2}=0
a=1 , b=\sqrt[3]2+1 , c=\sqrt[3]{4}+\sqrt[3]{2}
{\Delta=b^2-4ac=(\sqrt[3]2+1)^2-4*1*(\sqrt[3]{4}+\sqrt[3]{2})=\sqrt[3]{4}+2\sqrt[3]{2}+1-4\sqrt[3]4-4\sqrt[3]2\approx-6,3<0}
brak pierwiastków
Zastosowane wzory skróconego mnożenia
a^3-b^3=(a+b)(a^2+ab+b^2) różnica sześcianów
a^2-b^2=(a-b)(a+b) różnica kwadratów