rozłóż na czynniki wielomiany

[newton_123]

1.\(\displaystyle{ W(x)=x ^3(x^2-7)^2-36x}\)
2.\(\displaystyle{ W(x)=(x^2-2)^4-4x^4}\)

[andkom]

1.
\(\displaystyle{ x^3(x^2-7)^2-36x=x[(x^3-7x)^2-6^2]=x(x^3-7x-6)(x^3-7x+6)=\\
=x(x+1)(x^2-x-6)(x-1)(x^2+x-6)=\\=x(x+1)(x-3)(x+2)(x-1)(x+3)(x-2)=\\
=(x-3)(x-2)(x-1)x(x+1)(x+2)(x+3)}\)

2.
\(\displaystyle{ (x^2-2)^4-4x^4=[(x^2-2)^2+2x^2][(x^2-2)^2-2x^2]=\\ =[(x^2-2)^2+2x^2][(x^2-2)+\sqrt2x][(x^2-2)-\sqrt2x]=\\ =[(x^4-4x^2+4)+2x^2](x^2+\sqrt2x-2)(x^2-\sqrt2x-2)=\\ =(x^4-2x^2+4)(x^2+\sqrt2x-2)(x^2-\sqrt2x-2)=\\ =[(x^4+4x^2+4)-6x^2](x^2+\sqrt2x-2)(x^2-\sqrt2x-2)=\\ =[(x^2+2)^2-(\sqrt6x)^2](x^2+\sqrt2x-2)(x^2-\sqrt2x-2)=\\ =[(x^2+2)-\sqrt6x][(x^2+2)+\sqrt6x](x^2+\sqrt2x-2)(x^2-\sqrt2x-2)=\\ =(x^2-\sqrt6x+2)(x^2+\sqrt6x+2)(x^2+\sqrt2x-2)(x^2-\sqrt2x-2)}\)
Dwa ostatnie czynniki można jeszcze dalej rozkłać.

[newton_123]

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