Znajdź Pierwiastki Wielomianu

[Plebansk8]

a) \(\displaystyle{ x^{4}+2x^{3}-12x^{2}-18x+27}\)
b) \(\displaystyle{ 2x^{3}-9x^{2}-6x+5}\)
c) \(\displaystyle{ x^{4}-5x^{3}+3x^{2}+15x-18}\)
Prosze o pomoc

[sea_of_tears]

\(\displaystyle{ x^4+2x^3-12x^2-18x+27=\newline
=x^4-x^3+3x^3-3x^2-9x^2+9x-27x+27=\newline
=x^3(x-1)+3x^2(x-1)-9x(x-1)-27(x-1)=\newline
=(x-1)(x^3+3x^2-9x-27)=\newline
=(x-1)[x^2(x+3)-9(x+3)]=\newline
=(x-1)(x+3)(x^2-9)=(x-1)(x+3)(x-3)(x+3)=(x-1)(x-3)(x+3)^2\newline
x=1, x=3, x=-3}\)

[ Dodano: 26 Listopada 2008, 23:03 ]
\(\displaystyle{ 2x^3-9x^2-6x+5=\newline
=2x^2+2x^2-11x^2-11x+5x+5=\newline
=2x^2(x+1)-11x(x+1)+5(x+1)=\newline
=(x+1)(2x^2-11x+5)\newline
x=-1\newline
\Delta=121-40=81\newline
x_2=\frac{11-9}{4}=\frac{1}{2}\newline
x_3=\frac{11+9}{4}=5}\)

[ Dodano: 26 Listopada 2008, 23:10 ]
\(\displaystyle{ x^4-5x^3+3x^2+15x-18=\newline
=x^4-2x^3-3x^3+6x^2-3x^2+6x+9x-18=\newline
=x^3(x-2)-3x^2(x-2)-3x(x-2)+9(x-2)=\newline
=(x-2)(x^3-3x^2-3x+9)=\newline
=(x-2)[x^2(x-3)-3(x-3)]=\newline
=(x-2)(x-3)(x^2-3)=\newline
=(x-2)(x-3)(x-\sqrt3)(x+\sqrt3)\newline
x=2, x=3, x=\sqrt3, x=-\sqrt3}\)

[Plebansk8]

pięknie dziękuję !!