Question

The value of the expression $x^4-8x^3+18x^2-8x+2$, when $x=2+\sqrt{3}$, is:

• A. $2$
• B. $1$
• C. $0$
• D. $3$

Answer 1

Answer: (B)

$1$

Let $f\left(x\right)=x^4-8x^3+18x^2-8x+2$
Substituting $x=2+\sqrt{3}$, we get
$f(2+\sqrt{3})={(2+\sqrt{3})}^4-8{(2+\sqrt{3})}^3+18{(2+\sqrt{3})}^2-8(2+\sqrt{3})+2$

$={(2+\sqrt{3})}^2({(2+\sqrt{3})}^2+18)-8(2+\sqrt{3})({(2+\sqrt{3})}^2+1)+2$

$=(7+4\sqrt{3})(4\sqrt{3}+25)-8(2+\sqrt{3})(8+4\sqrt{3})+2$

$=175+48-128-96+2$
$=1$
Hence, option B is correct.