Question

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

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:x^4\:-\:x^3\:-\:2x^2\:-\:3x\:+\:1\:=\:2\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:x\:=\:1\:+\:\sqrt{2}}$

We have to find the value of

$\displaystyle{\sf\:x^4\:-\:x^3\:-\:2x^2\:-\:3x\:+\:1}$

Now,

$\displaystyle{\sf\:x^4\:-\:x^3\:-\:2x^2\:-\:3x\:+\:1}$

$\displaystyle{\implies\sf\:x\:(\:x^3\:-\:x^2\:-\:2x\:-\:3\:)\:+\:1}$

$\displaystyle{\implies\sf\:x\:[\:x\:(\:x^2\:-\:x\:-\:2\:-\:3\:)\:]\:+\:1}$

$\displaystyle{\implies\sf\:x\:[\:x\:(\:x^2\:-\:2x\:+\:x\:-\:2\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:x\:[\:x\:(\:x\:-\:2\:)\:+\:1\:(\:x\:-\:2\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:x\:[\:(\:x\:-\:2\:)\:(\:x\:+\:1\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:[\:(\:1\:+\:\sqrt{2}\:)\:(\:1\:+\:\sqrt{2}\:-\:2\:)\:(\:1\:+\:\sqrt{2}\:+\:1\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:[\:(\:1\:+\:\sqrt{2}\:)\:(\:-1\:+\:\sqrt{2}\:)\:(\:2\:+\:\sqrt{2}\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:[\:(\:\sqrt{2}\:-\:1\:)\:(\:\sqrt{2}\:+\:1\:)\:(\:\sqrt{2}\:+\:2\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:[\:(\:\sqrt{2}^2\:-\:1^2\:)\:(\:\sqrt{2}\:+\:2\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:[\:(\:2\:-\:1\:)\:(\:2\:+\:\sqrt{2}\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:[\:1\:\times\:(\:2\:+\:\sqrt{2}\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:[\:(\:2\:+\:\sqrt{2}\:)\:-\:3\:]\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:(\:2\:+\:\sqrt{2}\:-\:3\:)\:+\:1}$

$\displaystyle{\implies\sf\:(\:1\:+\:\sqrt{2}\:)\:(\:-\:1\:+\:\sqrt{2}\:)\:+\:1}$

$\displaystyle{\implies\sf\:(\:\sqrt{2}\:)^2\:-\:1^2\:+\:1}$

$\displaystyle{\implies\sf\:2\:-\:1\:+\:1}$

$\displaystyle{\implies\sf\:2\:+\:0}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:x^4\:-\:x^3\:-\:2x^2\:-\:3x\:+\:1\:=\:2\:}}}}$

Answer 2

x⁴ - x³ - 2x² - 3x + 1

x = 1+√2

x² + x² - x³ - 2x² - 3x + 1

after simplifying ,

3x² - 4x + 1

then ,

3 + 6 - 4 - 4√2 + 1 = 6 - 4√2