Question

If $x=1+\sqrt{2}$ then find the value of the expression $x^4-x^3-2x^2-3x+1$

Answer 1

$x=1+\sqrt{2}$

$x^4-x^3-2x^2-3x+1$

$\Rightarrow x^2(x^2-x-2)-3x+1$

$\Rightarrow x^2(x-2)(x+1)-3x+1$

$\Rightarrow {(\sqrt{2}+1)}^2(\sqrt{2}+1-2)(\sqrt{2}+1+1)-3x+1$

$\Rightarrow (\sqrt{2}+1)(\sqrt{2}+1)(\sqrt{2}-1)(\sqrt{2}+2)-3x+1$

$\Rightarrow (\sqrt{2}+1)(2-1)(\sqrt{2}+2)-3x+1$

$\Rightarrow (\sqrt{2}+1)(\sqrt{2}+2)-3(\sqrt{2}+1)+1\Rightarrow 2+3\sqrt{2}+2-3\sqrt{2}-3+1$

$\Rightarrow 2$