Find roots of polynomial $$ p(x) = 1+x+x^2 $$
Answer
The roots of polynomial $ p(x) $ are:
$$ \begin{aligned}x_1 &= -\frac{ 1 }{ 2 }+\frac{\sqrt{ 3 }}{ 2 }i\\[1 em]x_2 &= -\frac{ 1 }{ 2 }- \frac{\sqrt{ 3 }}{ 2 }i \end{aligned} $$Explanation
Step 1:
Write polynomial in descending order
$$ \begin{aligned} 1+x+x^2 & = 0\\[1 em] x^2+x+1 & = 0 \end{aligned} $$Step 2:
The solutions of $ x^2+x+1 = 0 $ are: $ x = -\dfrac{ 1 }{ 2 }+\dfrac{\sqrt{ 3 }}{ 2 }i ~ \text{and} ~ x = -\dfrac{ 1 }{ 2 }-\dfrac{\sqrt{ 3 }}{ 2 }i$.
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