Question

Mostre que os polinômios f={x^2 +(raiz de 2)x+1} . {x^2 - (raiz de 2)x+1}=x^4+1

Answer 1

Olá!

$\\ \mathsf{(x^2 + x \sqrt{2} + 1) \cdot (x^2 - x \sqrt{2} + 1) =} \\\\ \mathsf{\left [ (x^2 + 1) + x\sqrt{2} \right ] \cdot \left [ (x^2 + 1) - x\sqrt{2} \right ] =} \\\\ \mathsf{(x^2 + 1)^2 - (x\sqrt{2})^2 =} \\\\ \mathsf{x^4 + 2x^2 + 1 - x^2 \cdot 2 =} \\\\ \mathsf{x^4 + 2x^2 + 1 - 2x^2 =} \\\\ \boxed{\mathsf{x^4 + 1}} \\\\ \blacksquare$

Answer 2

vamos lá...

$f=(x^2+x \sqrt{2} +1)(x^2-x \sqrt{2} +1)=4x^2+1 \\ \\ x^4-\not x^3 \sqrt{2} +x^2+\not x^3 \sqrt{2} -2x^2+\not x \sqrt{2} +x^2-\not x \sqrt{2} +1=4x^2+1 \\ \\ x^4+x^2-2x^2+x^2+1=4x^2+1 \\ \\ x^4+\not2x^2-\not2x^2+1=4x^2+1 \\ \\ 4x^2+1=4x^2+1$