Question : If $x+\frac{1}{x}=1$, then the value of $x^{12}+x^9+x^6+x^3+1$ is:
Option 1: 1
Option 2: - 1
Option 3: 0
Option 4: - 2
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Correct Answer: 1
Solution :
We have to find the value of $x^{12} + x^9 + x^6 + x^3 + 1$
Given, $x + \frac{1}{x} = 1$
$x^2 + 1 = x$
⇒ $x^2 - x + 1 = 0$
⇒ $x^3+1^3=(x + 1)(x^2 - x + 1) $
⇒ $(x^3 + 1^3) = 0$
⇒ $x^3 = -1$
According to the question,
$=x^{12} + x^9 + x^6 + x^3 + 1$
$=(x^3)^4 + (x^3)^3 + (x^3)^2 + (x^3) + 1$
$=(−1)^4 + (-1)^3 + (-1)^2 + (-1) + 1$
$=1 - 1 + 1 - 1 + 1$
$=3 - 2 = 1$
The value of $x^{12} + x^9 + x^6 + x^3 + 1$ is 1.
Hence, the correct answer is 1.
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