Question

Se considera expresia: E(x)=
${x}^{3} - 2 {x}^{2} - x + 2$
Aratati ca E(radical din 2)×E(-radical din 2) apartine N

Vreau doar sa ma verific.

Answer 1

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Answer 2

$E(x)=x^{3}-2x^{2}-x+2\\\\E(x)=x^{2}(x-2) -(x-2)\\\\E(x)=(x-2)(x^{2}-1)\\\\E(x)=(x-2)(x-1)(x+1)$

$E(\sqrt{2})=(\sqrt{2}-2)(\sqrt{2}^{2} -1)\\\\E(\sqrt{2})=(\sqrt{2} -2)(2-1)\\\\E(\sqrt{2})=(\sqrt{2}-2)$

$E(-\sqrt{2})=(-\sqrt{2}-2)[(-\sqrt{2})^{2} -1)\\\\E(-\sqrt{2})=(-\sqrt{2} -2)(2-1)\\\\E(-\sqrt{2})=(-\sqrt{2}-2)$

$E(\sqrt{2} )*E(-\sqrt{2})= (\sqrt{2}-2)(-\sqrt{2} -2)\\\\E(\sqrt{2} )*E(-\sqrt{2})=(-2)^{2}-(\sqrt{2})^{2}\\\\E(\sqrt{2} )*E(-\sqrt{2})=4-2\\\\E(\sqrt{2} )*E(-\sqrt{2})=2$ ∈ IN