First slide

Theory of equations

Question

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

Moderate

A

2
B

1
C

0
D

3

Solution

$x = 2 + \sqrt{3}$
or $(x - 2)^{2} = 3$
or $x^{2} - 4 x + 1 = 0$ (1)
or $(x - 2)^{4} = 9$
or $x^{4} - 8 x^{3} + 24 x^{2} - 32 x + 16 = 9$
or $x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2 + 6 (x^{2} - 4 x + 1) - 1 = 0$
Using (1), we get
$x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2 = 1$

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