Question : If $ a^{2}+a+1=0$, then the value of $a^{5}+a^{4}+1$ is:
Option 1: $a^2$
Option 2: $1$
Option 3: $0$
Option 4: $a+1$
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Correct Answer: $0$
Solution :
Given: $a^2 + a +1 = 0$
⇒ $a^2 + a = -1$
The given expression = $a^5 + a^4+1$
$=a^3(a^2 + a) + 1$
$=a^3(-1) + 1 = 1^3 - a^3$
Using identity $x^3-y^3=(x-y)(x^2+y^2+xy)$, we get
$=(1-a)(1^2+1(a)+a^2)$
$=(1-a)(1+a+a^2)$
$= 0$
Hence, the correct answer is $0$.
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